# Proof Set is Countable : Bolzano-Weierstrass Theorem

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Given S is a subset of R

Suppose S' (set of all accumulation points in S) = emptyset

Prove S is countable.

I think I am supposed to use the Bolzano-Weierstrass Theorem but I can't figure out how to apply it.

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#### Solution Preview

Bolzano-Weierstrass Theorem : Every bounded infinte set in R has an accumulation point.

We have another basic fact for countable set: the union of countable ...

#### Solution Summary

The Bolzano-Weierstrass Theorem is applied to proving a set is countable. The proof is concise.

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