# Real numbers/Bolzano Weierstraß/Fact/Proof

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Proof

Suppose that the sequence is bounded by

We define inductively an interval bisection, such that in all the intervals, there are infinitely many members of the sequence. The initial interval is . Suppose that the -th interval is already constructed. We consider the two halves

Al least in one of these, there are infinitely many members of the sequence, and we choose the interval as one half with infinitely many members. By this method, the lengths of the intervals are bisected, and so we have a sequence of nested intervals. As a subsequence, we choose arbitrary elements

with . This is possible, as each interval contains infinitely members. This subsequence converges due to exercise to the number determined by the nested intervals.