(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let K be a subset of R. Prove that if every sequence in K has an accumulation point, then K must be compact.

2. Relevant equations

I tried to proof it below. Am I on the right track?

3. The attempt at a solution

My intuition;

Let x_n be sequence in K whose accumulation point is x, then there is a sub-sequence.

x_n_k converges to x.

Since sub-sequence x_n_k converges to x, x_n_k is a Cauchy sequence.

We know that Cauchy sequences are bounded. Thus x_n is bounded. so K is bounded.

Since all accumulation points are in K so K is closed.

By Heine-Borel Thm, K is compact.

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# Homework Help: Compactness with accumulation points

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