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Here is the question:

Prove that if the sequence {s} has no convergent subsequence then {|s|} diverges to infinity.

To me, this seems so easy, but I'm having a really hard time putting it down in a rigorous manner.

My thoughts are:

every convergent sequence has a convergent subsequence (theorem in the book), so if there is no convergent subsequence, then the sequence itsself cannot converge either.

So the absolute value won't converge.

I tried to do it with contradiction as follows:

Suppose that |s| converges. Then it has a convergent subsequence. Since |s| is also a subsequence of {s}, the convergent subsequence of |s| lies inside {s}. So {s} has a convergent subsequence. Contradiction.

Any clarification will be greatly appreciated.

CC

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# Homework Help: Sequence proof

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