Word Count : 261

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Bolzano–Weierstrass theorem (sequence form): every bounded sequence in (\mathbb{R}^n) has a convergent subsequence. Equivalently for sets in (\mathbb{R}): every infinite bounded subset of (\mathbb{R}) has at least one accumulation (limit) point. Consider the set [ S=\left{\frac1m-\frac1n:;m,n\in\mathbb{N}\right}. ] First note (S) is bounded: for all (m,n) we have (-1<\frac1m-\frac1n<1), so (S\subset(-1,1)). Also (S) is infinite (there are infinitely many ordered pairs ((m,n)) producing distinct values), so __________ _____ ____ ___ ______.
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