Here's the statement I want to prove:

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of real numbers that converges to a real number $L$. Then, every subsequence $\{a_{n_k}\}_{k=1}^{\infty}$ converges to $L$.

Proof Attempt:

Let $\epsilon > 0$ be arbitrary but fixed. We are required to prove that:

$$\exists K \in \mathbb{N}: \forall k \geq K: |a_{n_k}-L| < \epsilon$$

We know that there exists an $N_0 \in \mathbb{N}$ such that:

$$\forall n \geq N_0: |a_n-L| < \epsilon$$

Since $\{n_k\}_{k=1}^{\infty}$ is a strictly increasing sequence of natural numbers, then:

$$\exists K \in \mathbb{N}: \forall k \geq K: n_k \geq N_0$$

$$\implies \exists K \in \mathbb{N}: \forall k \geq K: |a_{n_k}-L| < \epsilon$$

which is exactly the assertion that $\lim_{k \to \infty} (a_{n_k}) = L$. That proves the desired result.

Is the proof above correct? If it isn't, why? How can I fix it?

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    $\begingroup$ Looks good to me $\endgroup$ – QC_QAOA Jul 8 '20 at 15:26
  • $\begingroup$ Thank you so much! $\endgroup$ – Abhi Jul 8 '20 at 15:28
  • $\begingroup$ It's slightly faster if you make use of $n_k\ge k$ because it's a strictly increasing positive integer sequence. $\endgroup$ – Peter Foreman Jul 8 '20 at 15:33
  • $\begingroup$ Yeap, that's the approach that my book takes. I read its solution after getting confirmation that mine was correct. I don't really know how i'm supposed to think of quick and easy solutions like that lol. $\endgroup$ – Abhi Jul 8 '20 at 15:35
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    $\begingroup$ Here's another one to try: Suppose $a_n $ is a sequence such that every subsequence has a further subsequence that converges to $L$. Prove that $a_n \to L$. This is a surprisingly useful technical lemma. $\endgroup$ – copper.hat Jul 8 '20 at 16:11

Your proof is correct. In fact, you could use your proof to derive a method to find an explicit suitable $K$ for each $\epsilon$, for the subsequence, given a method for the sequence itself.

  • $\begingroup$ Nice, thanks so much. So, in essence, I've also derived an algorithm for choosing $K$ for each given $\epsilon$. That sounds pretty cool, would it be important in other things i'll see in Analysis? Also, I'll accept your answer as soon as possible. It's not letting me do it right now. $\endgroup$ – Abhi Jul 8 '20 at 15:29
  • $\begingroup$ I think it could matter if you were interested in speeds of convergence. But I'm not sure that happens so often with subsequences. $\endgroup$ – FiMePr Jul 8 '20 at 16:21
  • $\begingroup$ @AbhijeetVats But, I guess that the book in which you found your exercise deals with more general facts in topology and analysis. So you probably don't need to worry about speeds of convergence yet. $\endgroup$ – FiMePr Jul 8 '20 at 16:27
  • $\begingroup$ Yea, Mathematical Analysis by Bernd Schroder. Not sure if it deals with that stuff because I'm not too far in. I'll see if I can do a bit of extra reading on what you've mentioned. When you say "speeds of convergence", I guess you mean how fast something is converging? Like, if we know that something converges, it's also useful to know how fast it converges relative to something else? $\endgroup$ – Abhi Jul 8 '20 at 16:29
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    $\begingroup$ @AbhijeetVats Yes, that is what I mean. $\endgroup$ – FiMePr Jul 8 '20 at 16:46

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