### Quick description

The Bolzano-Weierstrass theorem asserts that every bounded sequence of real numbers has a convergent subsequence. More generally, it states that if is a closed bounded subset of then every sequence in has a subsequence that converges to a point in . This article is not so much about the statement, or its proof, but about how to use it in applications. As we shall see, one of the signs to look out for is any statement that has the form "For every there exists such that ," where is some bounded set and is some property of real numbers (or elements of ).

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