Real Analysis 7 | Cauchy Sequences and Completeness

1. Formula
   Observation

   The handwritten expression $(a_n)_{n\in\mathbb{N}}$ appears on the left side of the board.

2. Audio
   Observation

   The narrator refers to “sequences” and “the sequence members here.”

($a_n$)_{$n\in\mathbb{N}$}

A real sequence indexed by natural numbers.

$n \in \mathbb{N}$

1. Formula
   Observation

   The word “convergent” is written next to $(a_n)_{n\in\mathbb{N}}$.

2. Audio
   Observation

   The narrator says “namely convergent sequences.”

convergent

Property of a sequence having a limit in the ambient number system.

1. Formula
   Observation

   The red parenthetical text reads “(there is a limit $a = \lim_{n\to\infty} a_n$)”.

2. Audio
   Observation

   The narrator says there is a number $a$ such that the sequence members get arbitrarily close to this number $a$ eventually.

a

The limit of the sequence $(a_n)_{n\in\mathbb{N}}$.

$a \in \mathbb{R}$ (implied by the real-analysis context; not explicitly stated in this clip)

1. Formula
   Observation

   The formula $\lim_{n\to\infty} a_n$ is written in red.

$\lim_{n\to\infty} a_n$

Limit notation for the sequence $(a_n)$ as $n$ tends to infinity.

1. Audio
   Observation

   The narrator says “for all epsilon” and later “less than epsilon we choose at the beginning.”

2. Formula
   Observation

   The final displayed condition uses $\varepsilon>0$ and $|a_n-a_m|<\varepsilon$.

$\varepsilon$

An arbitrary positive tolerance used in the convergence/Cauchy definitions.

$\varepsilon>0$

1. Audio
   Observation

   The narrator says “there exists a capital N.”

2. Formula
   Observation

   The displayed quantifier line contains $\exists N\in\mathbb{N}$.

N

A threshold index after which the required closeness condition holds.

$N \in \mathbb{N}$

1. Audio
   Observation

   The narrator says “for all indices n greater than this N.”

2. Formula
   Observation

   The displayed Cauchy condition uses $\forall n,m\ge N$.

n

A sequence index.

1. Audio
   Observation

   The narrator says “for all indices called n and m afterwards.”

2. Formula
   Observation

   The displayed condition contains $m$ in $\forall n,m\ge N$ and $|a_n-a_m|$.

m

A second sequence index used to compare two later terms of the sequence.

1. Audio
   Observation

   The narrator states that the distance between $a$ and $a_n$ is less than the given epsilon.

1. The formal convergence inequality is spoken but not visibly written in the sampled frames.

|$a-a_n$|

Distance between the limit candidate $a$ and the sequence term $a_n$.

a,$a_n \in \mathbb{R}$ (implied)

1. Diagram
   Observation

   A horizontal number line is drawn with the label $\mathbb{R}$ at the right end.

2. Audio
   Observation

   The narrator says “let’s look at the number line again.”

$\mathbb{R}$

The real number line used as the visual setting for the sequence terms.

1. Diagram
   Observation

   Tick marks on the number line are labeled $a_1$, $a_2$, $a_3$, followed by ellipsis-like accumulation marks.

2. Audio
   Observation

   The narrator says “here we have a one, a two, a three, and so on.”

$a_1$, $a_2$, $a_3$, $\ldots$

Individual terms of the sequence plotted on the real number line.

1. Formula
   Observation

   The green bracketed expression above the number line reads $|a_n-a_m|<\varepsilon$.

2. Audio
   Observation

   The narrator says the distance between two sequence members is “the absolute value of a n minus a m.”

|$a_n-a_m$|

Distance between two sequence terms $a_n$ and $a_m$.

$a_n$,$a_m \in \mathbb{R}$ (implied)

1. Formula
   Observation

   The board writes $(a_n)_{n\in\mathbb{N}}$ convergent and adds “(there is a limit $a = \lim_{n\to\infty} a_n$)”.

2. Audio
   Observation

   The narrator explains that convergence means there is a number $a$ such that the sequence members get arbitrarily close to $a$ eventually, then gives the epsilon-N wording.

1. The full formal convergence inequality is audible but not visibly written in the sampled frames.

A sequence $(a_n)_{n\in\mathbb{N}}$ is called convergent when there exists a number $a$ such that the terms $a_n$ become arbitrarily close to $a$ eventually. In the video this is expressed verbally as: for every $\varepsilon>0$ there is an $N\in\mathbb{N}$ such that for all indices $n\ge N$, the distance between $a$ and $a_n$ is less than $\varepsilon$.

1. The definition is applied to a sequence $(a_n)_{n\in\mathbb{N}}$.

2. It requires the existence of a limit $a$.

3. The epsilon-N formulation is stated verbally, not fully written on screen.

1. Audio
   Observation

   The narrator introduces a “different idea” that does not need a number $a$ in the definition.

2. Diagram
   Observation

   A number line with points $a_1,a_2,a_3,\ldots$ accumulating toward a region is drawn.

3. Formula
   Observation

   The green annotation $|a_n-a_m|<\varepsilon$ is placed above the clustered part of the number line.

Instead of measuring distance to an external limit $a$, the video motivates a property where the sequence members themselves get closer and closer to each other. The key quantity is the pairwise distance $|a_n-a_m|$ between two later terms, and the desired behavior is that this distance can be made smaller than any prescribed positive $\varepsilon$ once both indices are sufficiently large.

1. This is presented as the motivating idea behind the formal definition.

2. It applies to pairs of sufficiently late sequence terms.

3. The clip does not yet state the full formal name “Cauchy sequence” on screen.

1. Convergent sequence

1. Formula
   Observation

   The bottom line is written progressively as $\forall \varepsilon>0\ \exists N\in\mathbb{N}\ \forall n,m\ge N:\ |a_n-a_m|$ and completed with $<\varepsilon$.

2. Audio
   Observation

   The narrator reads the same quantifier structure aloud.

1. The clip ends while the narrator is still speaking the final phrase, but the displayed formula is complete enough to identify the standard Cauchy condition.

The formal condition written on the board is: for every positive $\varepsilon$, there exists a natural number $N$ such that for all indices $n,m\ge N$, the distance between the corresponding sequence terms satisfies $|a_n-a_m|<\varepsilon$. This is the epsilon-N formulation of a Cauchy sequence, although the clip only reaches the condition itself and does not yet display the concluding label.

1. Applies to a sequence $(a_n)_{n\in\mathbb{N}}$.

2. Requires the condition to hold for every $\varepsilon>0$.

3. The threshold $N$ may depend on $\varepsilon$.

4. Both indices $n$ and $m$ must be at least $N$.

1. Convergent sequence
2. Cauchy intuition: terms get close to each other

1. Formula
   Observation

   $\forall \varepsilon > 0 \quad \exists N \in \mathbb{N} \quad \forall n$,$m \geq N$ : |$a_n - a_m$| < $\varepsilon$

2. Audio
   Observation

   And now a sequence with this property we call a Cauchy sequence.

A sequence is called a Cauchy sequence if for every positive epsilon, there exists a natural number N such that for all indices n and m greater than or equal to N, the absolute difference between the terms $a_n$ and $a_m$ is less than epsilon.

1. Sequence of real numbers

1. Formula
   Observation

   Cauchy sequence $\iff$ Convergent sequence

2. Audio
   Observation

   being a Cauchy sequence is equivalent to being a convergent sequence... And the other one is exactly the completeness axiom.

The completeness axiom states that for a sequence of real numbers, being a Cauchy sequence is equivalent to being a convergent sequence. This means there are no 'holes' in the complete real number line.

1. Applies to sequences of real numbers

1. Definition of Cauchy Sequence

1. Formula
   Observation

   If $M \subseteq \mathbb{R}$ is bounded from above, then $\sup M \in \mathbb{R}$ (exists). If $M \subseteq \mathbb{R}$ is bounded from below, then $\inf M \in \mathbb{R}$ (exists).

The Dedekind completeness axiom states that every non-empty subset of the real numbers that is bounded above has a least upper bound (supremum) in the real numbers, and every non-empty subset that is bounded below has a greatest lower bound (infimum) in the real numbers.

1. M is a subset of the real numbers

2. M is bounded from above (for supremum)

3. M is bounded from below (for infimum)

1. Audio
   Observation

   either it's still an upper bound like $b_1$ or not.

2. Formula
   Observation

   $c_1 \text{ is an upper bound for } M$

An upper bound for a set M is a real number u such that for all elements m in M, m is less than or equal to u.

1. u is a real number

2. M is a subset of the real numbers

1. Formula
   Observation

   $c_1$ := $\frac{1}{2}(a_1 + b_1)$

2. Audio
   Observation

   So we just calculate the sum and then we divide by two.

The midpoint between two real numbers a and b is calculated by taking their average, which is the sum of the two numbers divided by two.

1. a and b are real numbers

1. Audio
   Observation

   by doing this calculation we get closer to this point which is the supremum of M

2. Formula
   Observation

   Two cases: (1) $c_n \text{ is an upper bound for } M$ : $b_{n+1}$ := $c_n$, $a_{n+1} = a_n$; (2) $c_n \text{ is not an upper bound for } M$ : $\exists x \in M$ : $x > c_n$, $a_{n+1} = x$, $b_{n+1} = b_n$

A recursive procedure to approximate the supremum of a bounded set M. It starts with an element $a_1$ in M and an upper bound $b_1$. It repeatedly calculates the midpoint $c_n$. If $c_n$ is an upper bound, it becomes the new upper bound $b_{n+1}$. If not, an element x in M greater than $c_n$ is found and becomes the new lower bound $a_{n+1}$. This narrows the interval [$a_n$, $b_n$] containing the supremum.

1. M is a non-empty subset of $\mathbb{R}$

2. M is bounded from above

1. Dedekind Completeness Axiom
2. Upper Bound
3. Midpoint Formula

1. Audio
   Observation

   Spoken: 'the sequence b n is indeed a Cauchy sequence'.

2. Formula
   Observation

   Written: $\Rightarrow (b_n)_{n \in \mathbb{N}}$ is a Cauchy sequence.

A sequence where the distance between any two terms becomes arbitrarily small as their indices increase. In this proof, it is established by showing |$b_n - b_m$| $\le (1/2)^{n-1}$|$b_1 - a_1$|, which gets arbitrarily small.

1. The sequence must be defined on a metric space or real numbers.

1. Audio
   Observation

   Spoken: 'using the completeness axiom to show that b n is a convergent sequence'.

2. Formula
   Observation

   Written: $\Rightarrow (b_n)_{n \in \mathbb{N}}$ is a convergent sequence with limit Sup M.

The property of the real numbers stating that every Cauchy sequence converges to a limit within the real numbers. This is the final step used to conclude the existence of the supremum.

1. Applies specifically to the field of real numbers $\mathbb{R}$.

1. Cauchy Sequence

1. Audio
   Observation

   Spoken: 'If the sequence a n is monotonically decreasing... and if the sequence is also bounded from below... then we can finally conclude...'

2. Formula
   Observation

   Written: Important application: If ($a_n$)_{$n \in \mathbb{N}$} is monotonically decreasing ($a_{n+1} \le a_n$ for all n) and bounded from below (the set {$a_n$}_{$n \in \mathbb{N}$} has a lower bound), then: ($a_n$)_{$n \in \mathbb{N}$} is convergent.

1. The video cuts off before the speaker finishes the sentence 'then we can finally conclude', but the text on screen explicitly states the conclusion.

An important application of the completeness of real numbers. It states that any sequence that is monotonically decreasing and bounded from below must converge.

1. The sequence must be monotonically decreasing.

2. The sequence must be bounded from below.

1. Completeness Axiom

1. Audio
   Observation

   The narrator says the problem with the convergence definition is that you need to know the limit to show convergence, because the distance is measured to $a$.

To prove convergence using the standard limit-based definition, one must already know or exhibit the limit $a$, because the condition measures $|a-a_n|$ against $\varepsilon$.

1. The sequence is being tested for convergence by the usual epsilon-N definition.

2. The definition uses distance to a specified limit $a$.

Implicitly universal over sequences considered with the standard convergence definition.

1. Audio
   Observation

   The narrator says there is a different property a sequence can have which does not need such a number $a$ in the definition.

2. Formula
   Observation

   The board replaces the limit-distance idea with the pairwise expression $|a_n-a_m|<\varepsilon$.

There is a sequence property defined without reference to a pre-existing limit $a$: instead of comparing terms to $a$, one compares later terms to each other via $|a_n-a_m|$.

1. One considers a sequence $(a_n)_{n\in\mathbb{N}}$.

2. The desired description should not require knowing a limit in advance.

No explicit quantifiers are stated in the audio at the moment of introduction; the formal quantifiers appear later in the displayed definition.

1. Formula
   Observation

   If $M \subseteq \mathbb{R}$ is bounded from above, then $\sup M \in \mathbb{R}$ (exists)

2. Audio
   Observation

   If M is such a set and also bounded from above, then we know the supremum exists.

If a subset M of the real numbers is bounded from above, then its supremum exists as a real number.

1. $M \subseteq \mathbb{R}$

2. M is bounded from above

$\forall M \subseteq \mathbb{R}$

1. Formula
   Observation

   If $M \subseteq \mathbb{R}$ is bounded from below, then $\inf M \in \mathbb{R}$ (exists)

2. Audio
   Observation

   we have the same thing for the infimum as well. So if we have a set that is bounded from below, then the infimum exists as a real number.

If a subset M of the real numbers is bounded from below, then its infimum exists as a real number.

1. $M \subseteq \mathbb{R}$

2. M is bounded from below

$\forall M \subseteq \mathbb{R}$

1. Formula
   Observation

   Cauchy sequence $\iff$ Convergent sequence

For a sequence of real numbers, being a Cauchy sequence is equivalent to being a convergent sequence.

1. The sequence consists of real numbers.

For all sequences of real numbers.

1. Audio
   Observation

   which gives us an approximation of the supremum... So in general when we have $a_n$ and $b_n$ we define $c_n$... So this is a recursive definition which gives us two sequences.

The bisection procedure recursively defines two sequences, ($a_n$) and ($b_n$), which approximate the supremum of the set M.

1. M is a non-empty subset of $\mathbb{R}$

2. M is bounded from above

3. $a_1 \in M$

4. $b_1$ is an upper bound for M

There exist sequences ($a_n$) and ($b_n$).

1. Formula
   Observation

   Top of screen: If $M \subseteq \mathbb{R}$ is bounded from above, then sup $M \in \mathbb{R}$ (exists).

2. Audio
   Observation

   Spoken: 'knowing that the supremum and infimum always exists'.

If a subset M of the real numbers is bounded from above, then its supremum exists in the real numbers.

1. $M \subseteq \mathbb{R}$

2. M is bounded from above

$\forall M \subseteq \mathbb{R}$

1. Formula
   Observation

   Written: $\Rightarrow (b_n)_{n \in \mathbb{N}}$ is a Cauchy sequence.

2. Audio
   Observation

   Spoken: 'formalize the conclusion that b n is indeed a Cauchy sequence'.

The sequence of upper bounds $b_n$ constructed via the nested interval method is a Cauchy sequence.

1. $b_n$ is constructed such that $b_{n+1} = b_n$ or $b_{n+1} = c_n$

2. |$b_n - b_m$| $\le (1/2)^{n-1}$|$b_1 - a_1$|

$\forall m > n \in \mathbb{N}$

1. Audio
   Observation

   The speaker says, 'we use the existence of the supremum of this set.'

Every non-empty set of real numbers that is bounded from above has a supremum (least upper bound).

1. The set is non-empty.

2. The set is bounded from above.

For all sets $M \subseteq \mathbb{R}$

1. Audio
   Observation

   The speaker says, 'if we use the existence of the infimum, we get another criterion.'

Every non-empty set of real numbers that is bounded from below has an infimum (greatest lower bound).

1. The set is non-empty.

2. The set is bounded from below.

For all sets $M \subseteq \mathbb{R}$

1. Diagram
   Observation

   A number line is drawn with $a_1,a_2,a_3,\ldots$ accumulating toward a cluster.

2. Audio
   Observation

   The narrator says the sequence members accumulate, but instead of using the limit, the members themselves get closer and closer to each other.

3. Formula
   Observation

   The annotation $|a_n-a_m|<\varepsilon$ is added above the clustered region, followed by the full quantified statement at the bottom.

1. Expression

   $$$$
   Explanation

   Start with a sequence $(a_n)_{n\in\mathbb{N}}$ and its usual convergence picture on the real line.

   Justification

   The narrator explicitly recalls convergent sequences before introducing the new idea.

   Shown in the video
2. Expression

   $$a_1,\ a_2,\ a_3,\ldots$$
   Explanation

   Plot the sequence terms on a number line and observe that they accumulate near one region.

   Justification

   This is shown visually by tick marks labeled $a_1,a_2,a_3,\ldots$ clustering on the right side of the line.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Notice that the accumulation suggests a limit, but the goal is to describe the phenomenon without naming that limit.

   Justification

   The narrator says they do not want to use the limit to describe what happens there.

   Shown in the video
4. Expression

   $$|a_n-a_m|$$
   Explanation

   Replace distance-to-limit by distance between two sequence members.

   Justification

   The narrator states that what happens is that the sequence members themselves get closer and closer to each other.

   Shown in the video
5. Expression

   $$|a_n-a_m|<\varepsilon$$
   Explanation

   Require that this mutual distance can be made smaller than any chosen positive tolerance.

   Justification

   The green annotation above the number line writes exactly this inequality.

   Shown in the video
6. Expression

   $$\forall\varepsilon>0\ \exists N\in\mathbb{N}\ \forall n,m\ge N:\ |a_n-a_m|<\varepsilon$$
   Explanation

   Formalize the idea by quantifying over all positive $\varepsilon$, some threshold $N$, and all later indices $n,m\ge N$.

   Justification

   The bottom line is written step by step and read aloud in the same order.

   Shown in the video

The geometric observation that later terms cluster together is converted into the formal Cauchy condition based on pairwise distances rather than distance to a known limit.

1. Audio
   Observation

   In the first case, we just have a smaller upper bound... so we can just substitute $b_1$... Or to put it in other words, we introduce $b_2$ to be the new upper bound given by $c_1$. On the other hand, the left-hand side we don't have to change, we just stay at $a_1$.

2. Formula
   Observation

   (1) $c_1 \text{ is an upper bound for } M$ : $b_2$ := $c_1$, $a_2 = a_1$

3. Audio
   Observation

   Now as I said, here it is possible to find a point x in M which is larger than $c_1$. Then of course, this new larger point should be our new $a_2$... And then we don't need to change the right-hand side with $b_1$.

4. Formula
   Observation

   (2) $c_1 \text{ is not an upper bound for } M$ : $\exists x \in M$ : $x > c_1$, $a_2 = x$, $b_2 = b_1$

1. Expression

   $$c_1 := \frac{1}{2}(a_1 + b_1)$$
   Explanation

   Calculate the midpoint $c_1$ between the initial element $a_1$ and upper bound $b_1$.

   Justification

   Definition of midpoint.

   Shown in the video
2. Expression

   $$\text{Case 1: } c_1 \text{ is an upper bound for } M$$
   Explanation

   Check if the midpoint $c_1$ is an upper bound for the set M.

   Justification

   Logical branching based on the property of $c_1$.

   Shown in the video
3. Expression

   $$b_2 := c_1, \quad a_2 = a_1$$
   Explanation

   If $c_1$ is an upper bound, it is a tighter upper bound than $b_1$. Set it as the new upper bound $b_2$, and keep the lower bound $a_2$ as $a_1$.

   Justification

   To narrow the interval containing the supremum.

   Shown in the video
4. Expression

   $$\text{Case 2: } c_1 \text{ is not an upper bound for } M$$
   Explanation

   If $c_1$ is not an upper bound, there must exist an element in M greater than $c_1$.

   Justification

   Definition of an upper bound.

   Shown in the video
5. Expression

   $$\exists x \in M : x > c_1$$
   Explanation

   Find an element x in M that is strictly greater than $c_1$.

   Justification

   Existence guaranteed by $c_1$ not being an upper bound.

   Shown in the video
6. Expression

   $$a_2 = x, \quad b_2 = b_1$$
   Explanation

   Set this new element x as the new lower bound $a_2$, and keep the upper bound $b_2$ as $b_1$.

   Justification

   To narrow the interval containing the supremum.

   Shown in the video

The bisection step updates the interval [$a_n$, $b_n$] to [$a_{n+1}$, $b_{n+1}$] such that the supremum of M remains within the new interval, and the length of the interval is halved.

1. Formula
   Observation

   Written step-by-step: For $m > n$ : |$b_n - b_m$| $\le$ |$b_n - a_n$| $\le (1/2)^{n-1}$|$b_1 - a_1$|

2. Audio
   Observation

   Spoken explanation of substituting $a_n$ for $b_m$ and cutting the distance in half.

1. Expression

   $$|b_n - b_m|$$
   Explanation

   Start with the distance between two terms of the sequence $b_n$ for indices $m > n$.

   Justification

   Definition of distance in the sequence.

   Shown in the video
2. Expression

   $$\le |b_n - a_n|$$
   Explanation

   Substitute $a_n$ for $b_m$. Since $a_n$ is a lower bound and $b_m$ is an upper bound, replacing $b_m$ with $a_n$ increases or maintains the distance.

   Justification

   Properties of the nested intervals where $a_k \le b_j$ for all k, j.

   Shown in the video
3. Expression

   $$\le \left(\frac{1}{2}\right)^{n-1} |b_1 - a_1|$$
   Explanation

   By construction, the distance between $a_n$ and $b_n$ is halved at each step starting from the initial interval [$a_1$, $b_1$].

   Justification

   Nested interval construction using midpoints $c_n = \frac{1}{2}(a_n + b_n)$.

   Shown in the video
4. Expression

   $$\text{gets arbitrarily small}$$
   Explanation

   As n increases, ($1/2$)^{n-1} approaches 0, making the entire expression arbitrarily small.

   Justification

   Limit of a geometric sequence with ratio $1/2$.

   Shown in the video

The sequence ($b_n$)_{$n \in \mathbb{N}$} is a Cauchy sequence.

1. Formula
   Observation

   Written: $\Rightarrow (b_n)_{n \in \mathbb{N}}$ is a convergent sequence with limit Sup M.

2. Audio
   Observation

   Spoken: 'the only possibility for a limit then is the supremum of M'.

1. Expression

   $$(b_n)_{n \in \mathbb{N}} \text{ is a Cauchy sequence}$$
   Explanation

   Established in the previous derivation.

   Justification

   Cauchy estimate derivation.

   Shown in the video
2. Expression

   $$(b_n)_{n \in \mathbb{N}} \text{ is a convergent sequence}$$
   Explanation

   Apply the completeness axiom of the real numbers.

   Justification

   Completeness axiom states every Cauchy sequence in $\mathbb{R}$ converges.

   Shown in the video
3. Expression

   $$\lim_{n \to \infty} b_n = \sup M$$
   Explanation

   Since $b_n$ are upper bounds approximating the least upper bound, the limit must be the supremum.

   Justification

   Definition of supremum and the construction of $b_n$.

   Derived from the video

The sequence $b_n$ converges to $\sup M$, proving the existence of the supremum.

1. Formula
   Observation

   The board shows the implication from the inequality to being a Cauchy sequence, and then to being a convergent sequence.

1. Expression

   $$|b_n - b_m| \leq \left(\frac{1}{2}\right)^{n-1} |b_1 - a_1|$$
   Explanation

   The distance between terms $b_n$ and $b_m$ (for $m>n$) is bounded by a term that gets arbitrarily small.

   Justification

   Given condition on the board.

   Shown in the video
2. Expression

   $$(b_n)_{n\in\mathbb{N}} \text{ is a Cauchy sequence}$$
   Explanation

   Because the distance between terms can be made arbitrarily small, the sequence satisfies the definition of a Cauchy sequence.

   Justification

   Definition of a Cauchy sequence.

   Derived from the video
3. Expression

   $$(b_n)_{n\in\mathbb{N}} \text{ is a convergent sequence with limit } \sup M$$
   Explanation

   By the completeness of the real numbers, every Cauchy sequence converges to a limit, which in this context is the supremum of the set M.

   Justification

   Completeness axiom of real numbers.

   Derived from the video

The sequence $(b_n)$ is convergent with limit $\sup M$.

1. Diagram
   Observation

   Title card shows “The Bright Side of Mathematics” and “Real Analysis - Part 7”.

2. Audio
   Observation

   The narrator welcomes viewers back to Real Analysis and says today’s part is about Cauchy sequences and completeness.

1. Title text “The Bright Side of Mathematics”

2. Subtitle “Real Analysis - Part 7”

3. Steady membership banner

4. Channel logo

1. The static title card remains on screen while the narrator introduces the topic.

1. No mathematical formula is yet displayed during the first seconds.

This segment establishes the lecture context and announces the subject matter before any formal content appears.

1. Formula
   Observation

   $(a_n)_{n\in\mathbb{N}}$ appears, then the word “convergent”, then the red parenthetical “(there is a limit $a = \lim_{n\to\infty} a_n$)”.

2. Audio
   Observation

   The narrator recalls convergent sequences and explains the limit-based definition.

1. The formal convergence inequality is spoken but not visibly written in the sampled frames.