Bolzano–Weierstrass theorem

Suppose the sequence lies in $[A,B]$. Bisect $[A,B]$ into two halves; at least one half contains infinitely many terms of the sequence, so keep that half. Repeat, bisecting and keeping a half with infinitely many terms at each step, producing a nested sequence of intervals whose lengths shrink to $0$. By completeness these nested intervals shrink to a single point $L$. Picking one sequence term from each successive interval (with strictly increasing indices, always possible since each interval still contains infinitely many terms) yields a subsequence converging to $L$.