# Limit Proof, Absolute Value

Suppose $\displaystyle \lim_{n \rightarrow \infty}$$x_n = c. Prove that \displaystyle \lim_{n \rightarrow \infty} |x_n| = |c|. My gut tells me the triangle inequality is what I need to use, but I can't seem to reason it out. Thanks. - ## 1 Answer Yes, use the reverse triangular inequality http://en.wikipedia.org/wiki/Triangle_inequality:$$ ||x_n|-|c||\leq |x_n-c|.$$And then the squeeze theorem. - 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