a(sub n) >=0 for all n and has no convergent subsequence, show lim n->infinity a(sub n) = infinity?

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a(sub n) >=0 for all n and has no convergent subsequence, show lim n->infinity a(sub n) = infinity?

2007-09-25 18:33:46 · 2 answers · asked by mandy 1 in Science & Mathematics ➔ Mathematics

2 answers

If a sequence is convergent, then every subsequence is convergent with the same limit. Therefore, if a sequence has no convergent subsequence, it must be divergent. Since the terms are non-negative, the sequence must incrrease without bound, or limit = infinity.

2007-09-25 19:34:34 · answer #1 · answered by Tony 7 · 0⤊ 0⤋

That is Bolzano lemma
Proof here:
http://planetmath.org/?op=getobj&from=objects&id=2129

2007-09-26 02:35:05 · answer #2 · answered by Theta40 7 · 0⤊ 0⤋