Real numbers/Bolzano Weierstraß/Fact/Proof

Proof

Suppose that the sequence ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ is bounded by

${}a_{0}\leq x_{n}\leq b_{0}\,.$

We define inductively an interval bisection, such that in all the intervals, there are infinitely many members of the sequence. The initial interval is ${}I_{0}:=[a_{0},b_{0}]$. Suppose that the ${}k$-th interval ${}I_{k}$ is already constructed. We consider the two halves

$[a_{k},{\frac {a_{k}+b_{k}}{2}}]\,\,{\text{ and }}\,\,[{\frac {a_{k}+b_{k}}{2}},b_{k}].$

Al least in one of these, there are infinitely many members of the sequence, and we choose the interval ${}I_{k+1}$ 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

${}x_{n_{k}}\in I_{k}\,$

with ${}n_{k}>n_{k-1}$. This is possible, as each interval contains infinitely members. This subsequence converges due to exercise to the number ${}x$ determined by the nested intervals.

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