Sequence convergence

A sequence converges when its terms eventually lie within every prescribed distance of a limit.

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Different ways to understand it

Formal definition for limit of a sequence | Series | AP Calculus BC | Khan Academy
Khan Academy · English · Explanation
Connection evidence
Reviewed English material v3, knowledge item at 1:28–2:52: For every positive tolerance epsilon there is a positive threshold M; if $n>M$ , then | $a_n-L$ |<epsilon. The source states this quantified definition of sequence convergence.
Real Analysis | Sequences and the ε-N definition of convergence.
Michael Penn · English · Explanation
Connection evidence
Reviewed English material v1: card "ε-N definition of sequence convergence" at 4:00 — The video uses the standard definition shown on the left board: a sequence converges to L if for every $ε>0$ there is N∈ℕ such that | $a_n-L$ |<ε for all $n\ge N$ . In this example, $a_n=1/n^2$ and $L=0$ , so the goal is to make | $1/n^2-0$ | smaller than any prescribed positive tolerance.
Using the Monotone Convergence Theorem to Prove a Recursive Sequence Converges: Example 1
Mike, the Mathematician · English · Proof
Connection evidence
Reviewed proof connection to this video.

Questions and answers about Sequence convergence

How do you find the actual limit after knowing the sequence converges?

Once convergence to a limit $L$ is established, pass to the limit in the recurrence relation $a_{n+1}=\frac{1}{3}(a_n+1)$ . Since both $a_n$ and $a_{n+1}$ approach $L$ , substitute $L$ for both terms to get $L=\frac{1}{3}(L+1)$ , and then solve this linear equation for $L$ .

The sequence is known to converge to some real number $L$ .
The recurrence relation is $a_{n+1}=\frac{1}{3}(a_n+1)$ .

Example: The board shows the calculation: $L=\frac{1}{3}(L+1)\Rightarrow \frac{2}{3}L=\frac{1}{3}\Rightarrow L=\frac{1}{2}$ .

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Where is the Monotone Convergence Theorem used in this recursive-sequence proof?

The Monotone Convergence Theorem is used after establishing that the sequence is both bounded below and non-increasing. It is invoked to guarantee the existence of a real limit $L$ for the sequence, which is a prerequisite for solving the limit equation.

The sequence has been proven to be bounded from below.
The sequence has been proven to be non-increasing.

Example: The board writes “ $\{a_n\}_{n=1}^\infty$ bounded and non-increasing” and then “Monotone Convergence $\Rightarrow a_n\to L$ .”

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Why does the lecturer say the sequence must be bounded and monotone to use the Monotone Convergence Theorem?

The Monotone Convergence Theorem states that every bounded, monotone sequence of real numbers converges. Therefore, to prove convergence using this theorem, one must establish both boundedness and monotonicity as the necessary hypotheses.

The sequence consists of real numbers.
The goal is to prove convergence using the Monotone Convergence Theorem.

Example: The lecturer states they want to use the monotone convergence theorem and therefore need to show the sequence is bounded and non-decreasing or non-increasing.

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More concepts to explore

Cauchy sequences
Terms of a Cauchy sequence eventually become arbitrarily close to each other.
Completeness of the real numbers
Every nonempty real set bounded above has a least upper bound.
Bolzano–Weierstrass theorem
Every bounded sequence in finite-dimensional Euclidean space has a convergent subsequence.
Compactness
A space is compact when every open cover admits a finite subcover.
Epsilon–delta limits
Every prescribed output tolerance must follow from a sufficiently small nonzero input displacement.

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