Real Analysis 24 | Pointwise Convergence

1. Formula
   Observation

   ($f_1$, $f_2$, $f_3$, $f_4$, $f_5$, ...)

($f_1$, $f_2$, $f_3$, $f_4$, $f_5$, ...)

Sequence of functions

Set of functions indexed by natural numbers

1. Formula
   Observation

   $f_n$ : $I \rightarrow \mathbb{R}$

$f_n$

The n-th function in the sequence

Function from interval I to real numbers

1. Formula
   Observation

   $f_n$ : $I \rightarrow \mathbb{R}$

I

Domain of the functions in the sequence

Subset of real numbers (interval)

1. Formula
   Observation

   $\tilde{x}$

$\tilde{x}$

A fixed point in the domain I

Element of I

1. Formula
   Observation

   f : $I \rightarrow \mathbb{R}$

f

Limit function of the pointwise convergent sequence

Function from interval I to real numbers

1. Formula
   Observation

   $\forall \epsilon > 0$

$\epsilon$

Arbitrary positive real number for convergence tolerance

Positive real numbers

1. Formula
   Observation

   $\exists N \in \mathbb{N}$

N

Threshold index after which convergence holds

Natural numbers

1. Formula
   Observation

   $\forall n \geq N$

n

Index variable greater than or equal to N

Natural numbers

1. Formula
   Observation

   $f_n(x) = \frac{1}{n}x + 1$

$f_n(x)$

Specific example function for n-th term

Linear function on [0,1]

1. Formula
   Observation

   $f_n$ : [0,1] $\rightarrow \mathbb{R}$

[0,1]

Domain interval for the example sequence

Closed unit interval

1. Formula
   Observation

   The board repeatedly writes $f_n(x)$ and the sequence ($f_1$,$f_2$,$f_3$,$f_4$,$f_5$,...).

2. Audio
   Observation

   The lecturer refers to the sequence of functions and evaluates $f_n$ at a fixed point.

$f_n$

the n-th function in a sequence of functions

for the first example, $f_n$:[0,1]$\to\mathbb{R}$; for the second example, also $f_n$:[0,1]$\to\mathbb{R}$

1. Formula
   Observation

   The definition line writes f:I\to\mathbb{R} and later the conclusion writes f:[0,1]$\to\mathbb{R}$, $f(x)=1$.

2. Audio
   Observation

   The lecturer calls it the limit function and then the pointwise limit function f.

f

the pointwise limit function of the sequence ($f_n$)

in the definition, f:I\to\mathbb{R}; in the worked first example, f:[0,1]$\to\mathbb{R}$

1. Audio
   Observation

   And now such a sequence can have a convergence property we call pointwise convergence.

2. Formula
   Observation

   Pointwise convergence: ($f_1$, $f_2$, $f_3$, $f_4$, $f_5$, ...) is pointwisely convergent to a function f : $I \rightarrow \mathbb{R}$ if for all $\tilde{x} \in I$ : ($f_1(\tilde{x})$, $f_2(\tilde{x})$, $f_3(\tilde{x})$, $f_4(\tilde{x})$, $f_5(\tilde{x})$, ...) is convergent to $f(\tilde{x})$.

A sequence of functions ($f_n$) with domain I is pointwise convergent to a function f if, for every fixed point x̃ in I, the numerical sequence ($f_n$(x̃)) converges to f(x̃). This reduces function convergence to ordinary sequence convergence at each point.

1. Sequence of functions $f_n$ : I → ℝ

2. Limit function f : I → ℝ

3. For all x̃ ∈ I

4. For all $ε > 0$

5. Exists N ∈ ℕ

6. For all $n \ge N$

1. Sequence of Functions

1. Audio
   Observation

   Here please remind yourself that we have already learned in the last video what a sequence of functions is.

2. Formula
   Observation

   sequence of functions: ($f_1$, $f_2$, $f_3$, $f_4$, $f_5$, ...) $f_n$ : $I \rightarrow \mathbb{R}$

A sequence of functions is an ordered collection of functions $f_n$, each mapping from a common domain I to the real numbers ℝ, indexed by natural numbers n.

1. Common domain I

2. Codomain ℝ

3. Indexed by n ∈ ℕ

1. Formula
   Observation

   The board displays: Pointwise convergence: ($f_1$,$f_2$,$f_3$,$f_4$,$f_5$,...) is pointwisely convergent to a function f:I\to\mathbb{R} if for all $\tilde{x}\in I$: ($f_1(\tilde{x})$,$f_2(\tilde{x})$,$f_3(\tilde{x})$,$f_4(\tilde{x})$,$f_5(\tilde{x})$,...) is convergent to $f(\tilde{x})$.

2. Formula
   Observation

   The board also displays the epsilon form: [$\forall \tilde{x}\in I$\;$\forall \varepsilon>0$\;$\exists N\in\mathbb{N}$\;$\forall n\ge N$:\;|$f_n(\tilde{x})-f(\tilde{x})$|<$\varepsilon$].

3. Audio
   Observation

   The lecturer explains the idea by fixing a point $\tilde{x}$ and looking at the resulting numerical sequence.

A sequence of functions ($f_n$) converges pointwise to f on I exactly when, after fixing any single point $\tilde{x}\in I$, the numerical sequence ($f_n(\tilde{x})$) converges to $f(\tilde{x})$. The clip presents both the verbal definition and the epsilon formulation.

1. The sequence ($f_n$) and the limit function f are considered on a common domain I.

2. The quantifier order is important: $\tilde{x}$ is fixed first, then $\varepsilon$, then N may depend on both.

1. Formula
   Observation

   The board writes $\Rightarrow$ (pointwise) limit function f:[0,1]$\to\mathbb{R}$, $f(x)=1$.

2. Audio
   Observation

   The lecturer says: Hence our limit function is the constant function. To be more concrete, we should call it the pointwise limit function f.

When a sequence of functions converges pointwise, the function f obtained as the collection of pointwise limits is called the pointwise limit function. In the first example, that limit function is the constant function equal to 1 on [0,1].

1. Applies after pointwise convergence has been established on the whole domain.

1. Definition of pointwise convergence of a sequence of functions

1. Audio
   Observation

   The lecturer says one can immediately calculate this by just fixing a point x tilde and looking what happens.

2. Formula
   Observation

   The board writes For $\tilde{x}\in[0,1]$: $f_n(\tilde{x})=\frac{1}{n}\tilde{x}+1 \xrightarrow{n\to\infty} 1$.

To find the pointwise limit of ($f_n$), fix an arbitrary point $\tilde{x}$ in the domain, substitute it into $f_n$ to obtain a numerical sequence, compute its ordinary limit, and then repeat for all $\tilde{x}$.

1. The method applies point by point; the resulting N in the epsilon definition may depend on the chosen $\tilde{x}$.

1. Definition of pointwise convergence of a sequence of functions

1. Audio
   Observation

   The lecturer asks whether there is a pointwise limit function and then states the resulting limit functions explicitly.

2. Formula
   Observation

   The board writes '(pointwise) limit function f : [0,1] -> R' in both worked examples.

A sequence of functions ($f_n$) has a pointwise limit function f if, for each fixed input x in the domain, the numerical sequence $f_n(x)$ converges to $f(x)$. The video uses this definition directly in two examples on [0,1].

1. Each x is fixed first.

2. Convergence is checked separately for every x in the domain.

1. Audio
   Observation

   The lecturer says the sequence of numbers $f_n(x)$ is constant eventually, therefore it is convergent with limit 0.

2. Formula
   Observation

   The board shows $f_n(x)=0$ for all $n>1/x$ in the second case.

If for a fixed x the values $f_n(x)$ become equal to the same number for all sufficiently large n, then the numerical sequence $f_n(x)$ converges to that number. In the second example the eventual constant value is 0.

1. There exists an index threshold after which all terms are equal.

1. Pointwise limit function of a sequence of functions

1. Formula
   Observation

   The board writes $f_n$:[0,1]->R, $f_n(x)= n^2 x$(1-nx) for x in [0,$1/n$], and 0 for x in ($1/n$,1].

The second worked example is a sequence of piecewise functions on [0,1] whose nonzero part lives only on [0,$1/n$] and is given by a quadratic expression in x.

1. Defined on [0,1].

2. Index n is natural.

1. Formula
   Observation

   The board explicitly labels the conclusion as a '(pointwise) limit function' and writes f:[0,1]$\to\mathbb{R}$, $f(x)=0$.

In the worked example, the pointwise limit is obtained by fixing each x and checking the eventual value of $f_n(x)$. For $x=0$ the board shows $f_n(0)=0$ for all n, and for $x>0$ it shows $f_n(x)=0$ once $n>1/x$. Therefore the limiting function is identically zero on [0,1].

1. x is fixed while n varies.

2. The argument is split into the cases $x=0$ and $x>0$.

1. Audio
   Observation

   The narrator says, 'This is what we call the uniform convergence.'

1. The clip does not provide the formal definition of uniform convergence; only the name is spoken here.

At the start of this excerpt, the speaker names the topic just discussed as uniform convergence. The actual definition is deferred to the next video.

1. Only the term is introduced in this clip.

2. No formal criterion is stated here.

1. Audio
   Observation

   The narrator says, 'Now the definition and examples for the uniform convergence we will do in the next video.'

The speaker explicitly postpones the definition and examples of uniform convergence to the next video, so this clip functions as a transition from the current worked pointwise-limit example to the upcoming topic.

1. Applies to the structure of the lecture series rather than to a theorem statement.

1. Formula
   Observation

   The board shows $f_n(x)=\frac{1}{n}x+1$ and then $f_n(\tilde{x})=\frac{1}{n}\tilde{x}+1 \xrightarrow{n\to\infty} 1$.

2. Audio
   Observation

   The lecturer says this is one over n, a sequence convergent to zero, hence by our limit theorems the whole sequence goes to one, no matter what x tilde is.

For $f_n$:[0,1]$\to\mathbb{R}$ defined by $f_n(x)=\frac{1}{n}x+1$, the sequence converges pointwise to the constant function $f(x)=1$ on [0,1].

1. Each $f_n$ is defined on [0,1].

2. $f_n(x)=\frac{1}{n}x+1$.

For every fixed $\tilde{x}\in[0,1]$, $\lim_{n\to\infty} f_n(\tilde{x})=1$.

1. Formula
   Observation

   The board writes $f_n(\frac{1}{2}\cdot\frac{1}{n})=n^2\frac{1}{2n}(1-n\frac{1}{2n})=\frac{n}{4}$.

2. Audio
   Observation

   The lecturer says the maximum lies at one half times one over n and therefore the maximum here increases with n.

For the second example, the maximum of the parabolic branch occurs at $x=\frac{1}{2}\cdot\frac{1}{n}$ and equals $\frac{n}{4}$.

1. The first branch is $n^2x$(1-nx) on [0,$1/n$].

For each fixed n, the maximum on the nonzero branch is attained at $x=\frac{1}{2n}$ and has value $\frac{n}{4}$.

1. Audio
   Observation

   The lecturer concludes: 'Hence we have a pointwise limit function given by the zero function.'

2. Formula
   Observation

   The board writes '(pointwise) limit function f:[0,1]->R, $f(x)=0$' after splitting into $x=0$ and $x>0$.

For the sequence $f_n(x)=n^2 x$(1-nx) on [0,$1/n$] and 0 on ($1/n$,1], the pointwise limit on [0,1] is $f(x)=0$ for every x.

1. Each $f_n$ is defined on [0,1].

2. n ranges over the natural numbers.

For all $x \in [0,1]$.

1. Audio
   Observation

   The lecturer says pointwise convergence is maybe not strong enough for our needs, and that this motivates a stricter convergence property.

2. Diagram
   Observation

   The final picture shows continuous piecewise-linear functions converging pointwise to a step function with a jump.

Pointwise convergence alone may produce limit functions with qualitatively different behavior from the approximating sequence, such as jumps appearing in the limit even though every $f_n$ is continuous.

1. The sequence converges only pointwise.

2. No stronger mode of convergence is assumed.

Demonstrated by explicit examples.

1. Audio
   Observation

   The lecturer says the starting functions have no jumps, but the pointwise limit has a jump.

2. Diagram
   Observation

   The right-hand graph is a step function obtained as the pointwise limit of continuous piecewise-linear graphs.

A sequence of continuous functions can converge pointwise to a discontinuous function.

1. Each function in the sequence is continuous.

2. The convergence considered is pointwise.

Existential: there exists such a sequence.

1. Formula
   Observation

   The board writes: For $x=0$: $f_n(x)=0$ for all $n\in\mathbb{N}$; For $x>0$: $f_n(x)=0$ for all $n>1/x$; therefore (pointwise) limit function f:[0,1]$\to\mathbb{R}$, $f(x)=0$.

For the displayed sequence of functions on [0,1], the pointwise limit function is $f(x)=0$ for every $x\in[0,1]$.

1. The functions are considered on [0,1].

2. The case split $x=0$ and $x>0$ is used.

3. For $x>0$, one takes $n>1/x$.

For every fixed $x\in[0,1]$, eventually $f_n(x)=0$; hence the pointwise limit is 0 at each x.

1. Audio
   Observation

   Therefore it might not be so surprising that we can describe the whole thing efficiently by using quantifiers.

2. Formula
   Observation

   $\left[ \forall \tilde{x} \in I \quad \forall \epsilon > 0 \quad \exists N \in \mathbb{N} \quad \forall n \geq N : \quad |f_n(\tilde{x}) - f(\tilde{x})| < \epsilon \right]$

1. Expression

   $$\forall \tilde{x} \in I$$
   Explanation

   Fix an arbitrary point x̃ in the domain I

   Justification

   Pointwise convergence requires the property to hold for every point in the domain

   Shown in the video
2. Expression

   $$\forall \epsilon > 0$$
   Explanation

   Consider any positive tolerance ε

   Justification

   Standard epsilon-delta definition of sequence convergence

   Shown in the video
3. Expression

   $$\exists N \in \mathbb{N}$$
   Explanation

   There exists a threshold index N

   Justification

   Convergence guarantees such an N exists for the numerical sequence

   Shown in the video
4. Expression

   $$\forall n \geq N : |f_n(\tilde{x}) - f(\tilde{x})| < \epsilon$$
   Explanation

   For all indices $n \ge N$, the function values are within ε of the limit

   Justification

   This is the definition of convergence for the numerical sequence ($f_n$(x̃))

   Shown in the video

The pointwise convergence definition combines the universal quantifier over domain points with the standard epsilon-N convergence condition for each resulting numerical sequence.

1. Formula
   Observation

   The board successively writes For $\tilde{x}\in[0,1]$, then $f_n(\tilde{x})=\frac{1}{n}\tilde{x}+1$, then $\xrightarrow{n\to\infty}1$, then the limit function line.

2. Audio
   Observation

   The lecturer narrates the substitution, identifies the term $1/n$ as converging to 0, and concludes the whole sequence goes to 1.

1. Expression

   $$f_n(x)=\frac{1}{n}x+1$$
   Explanation

   Start from the given formula for the sequence of functions on [0,1].

   Justification

   Given on the board as the example definition.

   Shown in the video
2. Expression

   $$\text{Fix }\tilde{x}\in[0,1]$$
   Explanation

   Choose an arbitrary but fixed point in the domain to test pointwise convergence.

   Justification

   This is exactly the procedure stated in the definition of pointwise convergence.

   Shown in the video
3. Expression

   $$f_n(\tilde{x})=\frac{1}{n}\tilde{x}+1$$
   Explanation

   Substitute the fixed point into the formula to obtain a numerical sequence in n.

   Justification

   Direct substitution into $f_n(x)$.

   Shown in the video
4. Expression

   $$\frac{1}{n}\tilde{x}\to 0$$
   Explanation

   For fixed $\tilde{x}$, the factor $1/n$ tends to 0, so the product tends to 0.

   Justification

   Standard limit theorem for sequences multiplied by a fixed constant.

   Shown in the video
5. Expression

   $$f_n(\tilde{x})\to 1$$
   Explanation

   Adding the constant 1 gives the limit 1.

   Justification

   Limit laws for sums of convergent sequences.

   Shown in the video
6. Expression

   $$f(x)=1\quad\text{for all }x\in[0,1]$$
   Explanation

   Since the same argument works for every fixed point, the pointwise limit function is the constant 1.

   Justification

   The conclusion is independent of the particular choice of $\tilde{x}$.

   Shown in the video

The sequence $f_n(x)=x/n+1$ converges pointwise on [0,1] to the constant function $f(x)=1$.

1. Formula
   Observation

   The board writes $f_n(\frac{1}{2}\cdot\frac{1}{n})=n^2\frac{1}{2n}(1-n\frac{1}{2n})=\frac{n}{4}$.

2. Audio
   Observation

   The lecturer says we can immediately calculate this value, put this point into the function, cancel out a lot of n's, and get n over four.

1. Expression

   $$x_{\max}=\frac{1}{2}\cdot\frac{1}{n}$$
   Explanation

   The lecturer identifies the maximizing point of the parabolic branch.

   Justification

   Stated verbally and written on the board; the clip does not show a separate derivative computation.

   Shown in the video
2. Expression

   $$f_n\left(\frac{1}{2}\cdot\frac{1}{n}\right)=n^2\left(\frac{1}{2n}\right)\left(1-n\left(\frac{1}{2n}\right)\right)$$
   Explanation

   Substitute the maximizing x-value into the first branch $n^2x$(1-nx).

   Justification

   Direct substitution into the piecewise formula.

   Shown in the video
3. Expression

   $$=n^2\left(\frac{1}{2n}\right)\left(1-\frac{1}{2}\right)$$
   Explanation

   Simplify the inner product $n(1/(2n))$ to $1/2$.

   Justification

   Algebraic cancellation.

   Shown in the video
4. Expression

   $$=\frac{n}{4}$$
   Explanation

   Cancel powers of n and multiply the remaining constants.

   Justification

   Elementary algebra.

   Shown in the video

The maximum value of the nonzero branch is $n/4$, so it grows without bound as n increases.

1. Audio
   Observation

   The lecturer distinguishes the cases $x=0$ and $x>0$, then concludes the pointwise limit is 0.

2. Formula
   Observation

   The board writes 'For $x=0$: $f_n(x)=0$ for all $n \in N$' and 'For $x>0$: $f_n(x)=0$ for all $n>1/x$', followed by the limit function $f(x)=0$.

1. Expression

   $$f_n(x)=\begin{cases} n^2 x(1-nx), & x\in[0,1/n]\\ 0, & x\in(1/n,1]\end{cases}$$
   Explanation

   Start from the piecewise definition of the sequence on [0,1].

   Justification

   Given definition on the board.

   Shown in the video
2. Expression

   $$x=0 \Rightarrow f_n(0)=0 \quad \text{for all } n\in\mathbb{N}$$
   Explanation

   At the left endpoint the formula gives 0 regardless of n.

   Justification

   Direct substitution into the definition.

   Shown in the video
3. Expression

   $$x>0 \Rightarrow \exists N_x \text{ such that } n>N_x \implies x\in(1/n,1]$$
   Explanation

   Once n is larger than $1/x$, the fixed positive x lies in the zero branch of $f_n$.

   Justification

   Algebraic rearrangement of $n>1/x$.

   Shown in the video
4. Expression

   $$x>0 \Rightarrow f_n(x)=0 \quad \text{for all } n>1/x$$
   Explanation

   For all sufficiently large n, the value of the sequence at that fixed x is exactly 0.

   Justification

   Use of the second branch of the piecewise definition.

   Shown in the video
5. Expression

   $$\forall x\in[0,1],\; f_n(x) \text{ is eventually } 0$$
   Explanation

   Both cases show that for each fixed x the numerical sequence becomes constantly 0 after some index.

   Justification

   Combination of the $x=0$ and $x>0$ cases.

   Derived from the video
6. Expression

   $$f(x)=\lim_{n\to\infty} f_n(x)=0$$
   Explanation

   An eventually constant sequence converges to its constant tail value, so the pointwise limit is the zero function.

   Justification

   Definition of convergence of numerical sequences plus definition of pointwise limit.

   Shown in the video

The second example converges pointwise on [0,1] to $f(x)=0$.

1. Audio
   Observation

   The lecturer describes pushing the graph to the right, making the linear part steeper, and then taking the pointwise limit to get a function with a jump.

2. Diagram
   Observation

   The board shows several continuous piecewise-linear graphs and a separate step-function limit labeled 'pointwise limit'.

1. The exact algebraic formula for the third example is not written on screen in this clip.

1. Expression

   $$$$
   Explanation

   Begin with a continuous piecewise-linear function shaped like constant, then linear rise, then constant again.

   Justification

   Explicitly drawn on the board and described in audio.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Shift the whole graph slightly to the right to obtain the next function in the sequence.

   Justification

   Stated verbally by the lecturer.

   Shown in the video
3. Expression

   $$$$
   Explanation

   Continue the process so the linear transition region becomes steeper and narrower.

   Justification

   Stated verbally and visible in the successive curves.

   Shown in the video
4. Expression

   $$$$
   Explanation

   At each fixed x, the values eventually settle to one of the two constant levels, producing a step-shaped pointwise limit.

   Justification

   Inferred from the drawn limiting graph labeled 'pointwise limit'.

   Derived from the video

A sequence of continuous functions can converge pointwise to a discontinuous step function.

1. Formula
   Observation

   The board presents the two-case argument and concludes with the pointwise limit function.

1. Expression

   $$x=0:\quad f_n(x)=0\ \text{for all}\ n\in\mathbb{N}$$
   Explanation

   At the left endpoint, the value is already zero for every index n.

   Justification

   Directly written on the board.

   Shown in the video
2. Expression

   $$x>0:\quad f_n(x)=0\ \text{for all}\ n>\frac{1}{x}$$
   Explanation

   For any positive x, once n exceeds $1/x$, the corresponding function value is zero.

   Justification

   Directly written on the board.

   Shown in the video
3. Expression

   $$f:[0,1]\to\mathbb{R},\quad f(x)=0$$
   Explanation

   Combining the two cases shows that at each fixed x the sequence eventually equals 0, so the pointwise limit is the zero function.

   Justification

   Explicit conclusion labeled '(pointwise) limit function' on the board.

   Shown in the video

The pointwise limit function on [0,1] is identically zero.

1. Formula
   Observation

   The board shows $f_n(1/n)=n\cdot\frac{1}{2n}\left(1-\frac{1}{2n}\right)=\frac{n}{4}$.

1. The final equality is visually ambiguous in some frames: it can be read as $n/4$ or $n/4$!; the surrounding algebra supports $n/4$, but the handwriting is not perfectly clear.

1. Expression

   $$f_n\left(\frac{1}{n}\right)=n\cdot\frac{1}{2n}\left(1-\frac{1}{2n}\right)$$
   Explanation

   The board substitutes $x=1/n$ into the displayed formula for $f_n$.

   Justification

   Visible substitution line on the board.

   Shown in the video
2. Expression

   $$=\frac{n}{4}$$
   Explanation

   The displayed simplification gives a value growing like $n/4$ at $x=1/n$.

   Justification

   Visible final equality on the board, though the handwriting is slightly ambiguous.

   Shown in the video

At $x=1/n$, the displayed computation yields approximately $n/4$, indicating a peak value that grows with n.

1. Audio
   Observation

   So I would say let's look at an example. There we just need to look at a sequence of functions. Here the domain I should be just the interval zero to one. And the function $f_n$ should be defined as one over n times x plus one.

2. Formula
   Observation

   Example: $f_n$ : [0,1] $\rightarrow \mathbb{R}$, $\quad f_n(x) = \frac{1}{n}x + 1$

Consider the sequence of functions $f_n$ : [0,1] → ℝ defined by $f_n(x) = (1/n)x + 1$