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I have a proposition on the book Elementary Classical Analysis which states the following:

Let $x_n$ be a sequence in $\Bbb R$ and let x $\in \Bbb R$. Then $x_n \to x$ iff every subsequence of $x_n$ converges to $x$.

Could you please help me on the proof of this? I can't see any way to approach it.

1 Answers 1

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We need to prove two directions.

  1. When $x_n \rightarrow x$, can you use the limit definition to show that every subsequence must also converge to $x$? The limit definition, just to recall, is that for every $\epsilon > 0$, $|x_n - x|< \epsilon$ for all $n$ exceeding some $N$ depending only on $\epsilon$.

  2. If $x_n$ does not converge to $x$, can you see why at least one subsequence does not converge to $x$?

If you have any further questions feel free to comment.

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    I have troubles on such proofs because I tend to think on examples when question is this. I think of some subsequences that converge to x, then I got lost trying to generalize proof. Is this valid way to approach such question?2012-10-17