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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.


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up vote 2 down vote accepted

Yes, use the reverse triangular inequality $$ ||x_n|-|c||\leq |x_n-c|. $$

And then the squeeze theorem.

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I'm sorry could you elaborate on how the squeeze theorem might follow from here? – Peej Gerard Feb 17 '13 at 21:26
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. – 1015 Feb 17 '13 at 21:27
oh, great thanks, i think I follow you now – Peej Gerard Feb 17 '13 at 21:28
ah yes, fully clear to me now. Much appreciated! – Peej Gerard Feb 17 '13 at 21:30
@PaulGerard Great, then. – 1015 Feb 17 '13 at 21:30

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