Compactness with accumulation points

[sazanda]

Homework Statement

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

Homework Equations

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

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.

 

[PAllen]

Not quite. I think you're better off showing the equivalent (each implies the other):

if K is not compact, there exists a sequence without an accumulation point.

 

[Fredrik]

x_n_k is a Cauchy sequence.
We know that Cauchy sequences are bounded. Thus x_n is bounded.

Here you appear to be saying that if a sequence has a bounded subsequence, the sequence is bounded. Consider 1/2, 2, 1/3, 3, 1/4, 4, ...