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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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1 Answer 1

up vote 2 down vote accepted

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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I'm sorry could you elaborate on how the squeeze theorem might follow from here? –  Peej Gerard Feb 17 '13 at 21:26
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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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