Real numbers/Bolzano Weierstraß/Fact/Proof

Proof

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

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

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

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

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

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

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