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Suppose $\displaystyle \lim_{n \rightarrow \infty}$$x_n$ = $c$. Prove that $\displaystyle \lim_{n \rightarrow \infty}$ $|x_n|$ = $|c|$.

My gut tells me the triangle inequality is what I need to use, but I can't seem to reason it out.

Thanks.

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Yes, use the reverse triangular inequality http://en.wikipedia.org/wiki/Triangle_inequality: $$ ||x_n|-|c||\leq |x_n-c|. $$

And then the squeeze theorem.

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  • $\begingroup$ I'm sorry could you elaborate on how the squeeze theorem might follow from here? $\endgroup$ – Peej Gerard Feb 17 '13 at 21:26
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    $\begingroup$ By assumption, $\lim x_n=c$, ie $\lim (x_n-c)=0$ ie $\lim |x_n-c|=0$. So the rhs tends to $0$, and the lhs is bounded below by $0$. Squeeze indeed. $\endgroup$ – Julien Feb 17 '13 at 21:27
  • $\begingroup$ oh, great thanks, i think I follow you now $\endgroup$ – Peej Gerard Feb 17 '13 at 21:28
  • $\begingroup$ ah yes, fully clear to me now. Much appreciated! $\endgroup$ – Peej Gerard Feb 17 '13 at 21:30
  • $\begingroup$ @PaulGerard Great, then. $\endgroup$ – Julien Feb 17 '13 at 21:30

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