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If $\{x_{n}\}$ is a sequence of positive real numbers, $0 and $\lim_{n\to\infty}x_{n}=0$. We Know that any subsequence of $x_{n}$ will converges to zero, right! Now my question is: Can we find (construct) a subsequence $x'_{n}$ of $x_{n}$ such that $$\lim_{n\to\infty}\frac{x'_{n}}{x_{n}}=x$$ for nonzero $x$.

(For example, if $x_{n}=\frac{1}{n}$, then we can choose $x'_{n}:=x_{2n}=\frac{1}{2n}$ and we get $\lim_{n\to\infty}\frac{x'_{n}}{x_{n}}=1/2$).

Edit: Above I said "for nonzero $x$", and I didn't specified a value for $x$, all I want is just a nonzero limit.

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    True, any subsequence of a convergent sequence will also converge to the same limit. I suspect that the answer to your second question is not in general. There is a rather special uniformity to the example you chose, both in its monotonicity and the way it progresses. I'll think on a counterexample (unless someone beats me to it or comes up with a proof in the meantime).2012-06-10
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    Ok, as a summary: The result could be true if the sequence $x_{n}$ is increasing or decreasing, right!2012-06-11

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