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Subsequence

Suppose be a sequence and be an increasing sequence. Then is a subsequence of .

Existence of subsequence

Every sequence has a monotonic subsequence.

Bolzano-Weierstrass

Every bounded sequence on has a converging subsequence.

Theorem 1

Suppose is a sequence converging to , and is a subsequence of . Then is converging to .

Theorem 2

Suppose is a sequence diverging to , and is a subsequence of . Then is diverging to .

Subsequence of a cauchy sequence

If is Cauchy and is a subsequence converging to , then converges to .