2

In a metric space, a cluster point of a sequence $\{x_n\}$ is the limit of some subsequence. The only proof that I know works like this:

Construct a sequence $\delta _k \to 0$. For each $\delta _k$ find a point $x_{n_k}$ in the sequence which is within $\delta_k$ of the cluster point. Make sure that $n_{k} \lt n_{k+1}$, so that you really have a subsequence, and you can do this because there are only finitely many terms before index $n_{k}$.

Question: Is this using Countable Choice? If so is it because we're "picking" a sequence element for each $\delta_k$? If we are using Countable Choice, then is there a way to prove this without using CC?

I apologize if this is a duplicate question - I know there have been previous posts where it's explained why Sequential Continuity Implies $\delta - \epsilon$ Continuity requires Countable Choice. The argument used in that proof seems similar to this one.

Thank you very much.

bryanj
  • 3,938

1 Answers1

2

Suppose that $p$ is a cluster point of $\sigma=\langle x_n:n\in\Bbb N\rangle$. For each $k\in\Bbb N$ let

$$n_k=\min\{n\in\Bbb N:d(p,x_n)<2^{-n}\text{ and }\forall i<k(n_i<n)\}\;;\tag{1}$$

then $\langle x_{n_k}:k\in\Bbb N\rangle$ is a subsequence of $\sigma$ converging to $p$. The definition in $(1)$ does not require choice: it uniquely specifies $n_k$, and the hypothesis ensures that the set over which the minimum is taken is non-empty.

Brian M. Scott
  • 616,228
  • 1
    You typed faster than me. :-) – Andrés E. Caicedo Jul 05 '13 at 05:04
  • OK I think I got it. So the crucial difference between this situation and the sequential continuity situation is that we have this well-ordered index on the elements in the sequence, which allows us to exactly specify the subsequence elements? – bryanj Jul 05 '13 at 05:09
  • Yes. Also, note that rather than having to pick the $\delta_k\to0$, you can just work with the $1/2^n$. (So everything is rather explicit.) – Andrés E. Caicedo Jul 05 '13 at 05:16
  • Crystal clear. Thanks! – bryanj Jul 05 '13 at 05:21
  • @bryanj: You’re welcome! (And do note Andres’ point about being able to use a specific ‘measuring scale’ to get convergence rather than having to pick the $\delta_k$’s: I meant to mention that myself and forgot.) – Brian M. Scott Jul 05 '13 at 05:23
  • 1
    @Andres: And often (e.g. when I'm awake), I am told by Brian that I type faster than him. But many times I feel that you type faster than me! – Asaf Karagila Jul 05 '13 at 06:54