Suppose $(s_n)_{n\geq1}$ is a Cauchy sequence of real numbers. There exists a real number $s$ such that $\lim_{n\to\infty} s_n=s$

Question

prove the following by using subsequences which is by Bolzano-Weierstrass theorem

Suppose ($s_n$) is a Cauchy sequence of real numbers. There exists a real number s such that lim n→∞$s_n$ = s

here some of my works

We have |$s_n$ −s| = |$s_n −s_nk + s($_nk)$ −s$| ≤ |$s_n −s_nk$|+|$s_nk −s$|. how we can choose n and $n_k$ s.t $n_k$>= k

could you please help me how can I complete it ?

Answer 1

Hint/Sketch: You already realized that Bolzano Weirstrass is relevant.

What are the hypotheses for the theorem to hold? You need boundedness of your sequence, so start by proving that Cauchy sequences are bounded.

Next, once you have convergence along a subsequence to some limit $s$, what does the Cauchy condition tell you about the other terms in the sequence and how they relate to this subsequence?