Given with , build nested intervals by bisection: , and each is a half of containing infinitely many terms of the sequence. Then , so by the nested interval property there is a unique point .

Choose indices inductively: pick with ; since contains infinitely many terms, pick with ; continuing gives a subsequence with for every . Hence , and since the interval lengths shrink to zero, as .

The Bolzano-Weierstrass theorem also holds for complex sequences.