To prove that a sequence converges, find one or more convergent subsequences

Quick description

To prove that a sequence converges, it is sometimes easier to start by finding a subsequence that converges (or proving that such a subsequence exists). Then one can hope to deduce that the sequence itself converges. Another way of using subsequences is to exploit the following result: if every subsequence has a further subsequence that converges to a limit $L$ , then the whole sequence converges to $L$ .

Prerequisites

Basic concepts of real analysis.

This article is incomplete. Some general discussion would be nice, as would an example of the second method of proving convergence.

Example 1

To prove that a Cauchy sequence $(x_n)$ of real numbers converges, assuming one of the other forms of the completeness axiom, one can argue as follows. First, one proves the Bolzano-Weierstrass theorem. Then one observes that a Cauchy sequence is bounded, and so by the Bolzano-Weierstrass theorem has a convergent subsequence. And finally, one proves that if a Cauchy sequence has a subsequence that tends to a limit $L$ , then the sequence itself tends to $L$ .

Example 2

Here is a second way of proving that Cauchy sequences converge. Let $(x_n)$ be a Cauchy sequence. By the Cauchy condition we can choose a subsequence $(x_{n_k})$ such that $|x_{n_k}-x_m|\leq 2^{-k}$ whenever $m\geq n_k$ . It follows that the closed intervals $[x_{n_k}-2\cdot 2^{-k},x_{n_k}+2\cdot 2^{-k}]$ are nested, since if $k<l$ then $x_{n_l}\in[x_{n_k}-2^{-k},x_{n_k}+2^{-k}]$ , and $2\cdot 2^{-l}<2^{-k}$ . Therefore, their intersection is non-empty. Since their lengths tend to zero, the intersection is a singleton $\{x\}$ , and it is easy to check that the subsequence $x_{n_k}$ converges to $x$ . Therefore, since $(x_n)$ is Cauchy, so does the whole sequence.

General discussion

The trick in the second approach was to replace a Cauchy sequence by a subsequence that was rapidly Cauchy. That gave us the inequality $2\cdot 2^{-l}<2^{-k}$ , which was crucial to the success of the proof.