# Rudin's proof of every bounded sequence has a convergent subsequence.

This may be really simple, but I couldn't prove the last detail needed in the proof, see below.

Theorem

Every bounded sequence $$s_n$$ contains a convergent subsequence.

Let $$E=\{s_n:n\in\Bbb N\}$$. Then $$E$$ is bounded. If $$E$$ is infinite, it has a limit (by a previous theorem). And the conclusion follows from theorem 3.6 $$[\dots]$$

Okay, theorem $$3.6$$ says that every set of complex numbers with a limit point $$z$$ has a sequence $$z_n\in E\,(n=1,2,\dots)$$ such that $$z_n\to z$$.

The thing is that just having a sequence of $$z_{n_k}\to z$$ doesn't necessarily imply that $$z_{n_k}$$ is a subsequence of $$s_n$$ ($$z_{n_k}$$ could be $$(s_5,s_2,s_1,s_3,\cdots)$$).

How can we construct a subsequence of $$s_n$$ from such sequence $$z_{n_k}$$?

• That the limit is z means that for every neighborhood of z there exists an N such that for all n > N the z_n are in that neighborhood of z. Apr 2 '16 at 23:55

Suppose that $\langle n_k:k\in\Bbb Z^+\rangle$ is any sequence of distinct positive integers; then it has a strictly increasing subsequence $\langle n_{i_j}:j\in\Bbb Z^+\rangle$. We simply construct the sequence $\langle i_j:j\in\Bbb Z^+\rangle$ recursively, so that both it and $\langle n_{i_j}:j\in\Bbb Z^+\rangle$ are strictly increasing.
Let $i_1=1$. Given $i_j$ for some $j\in\Bbb Z^+$, let
$$i_{j+1}=\min\{k\in\Bbb Z^+:k>i_j\text{ and }n_k>n_{i_j}\}\;;$$
it’s clear that this is always possible, since only finitely many terms $n_k$ are less than or equal to $n_{i_j}$. Thus, the construction goes through, and it’s clear that $\langle n_{i_j}:j\in\Bbb Z^+\rangle$ is as desired.
• What if $n_k$ is not a sequence of distinct positive integers (as I believe this hypothesis is not guaranteed from what I wrote). Apr 3 '16 at 0:09
• Awesome! Thank you Brian. I just realized that in the proof of theorem $3.6$, rudin constructs the sequence $z_k$ such that all the $z_k$ are distinct, thus $n_k$ is a sequence of distinct integers (so my question wasn't really necessary), anyway, it's nice to know what can be done if that detail is not remembered. Apr 3 '16 at 0:20
• @YoTengoUnLCD: You’re welcome. I should perhaps have said a little more, but you’re generally pretty well on top of things, so I let myself be a little lazy. :-) Apr 3 '16 at 0:21