# limit of absolute value of something equals absolute value of limit of something?

Does limit of absolute value of something always equal absolute value of limit of something? Specifically, can I just say the below equality is true or do I need to prove it? I'm not sure how to prove it. Could you help me?

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If $a_n=(-1)^n$ then $\lim a_n$ doesn't exist (so neither does $|\lim a_n|$, yet $\lim |a_n|=1.$ – coffeemath Aug 9 '13 at 5:59
@AWertheim: Your proposed counterexample looks suspicious; you've brought a summation outside the absolute value, while the OP only brought out a limit. :) – Andrew D. Hwang Aug 9 '13 at 11:34
@AndrewD.Hwang goodness, how did I miss that? Tired eyes! Thanks. :) – Alex Wertheim Aug 9 '13 at 14:58

Hint: $f(x) = |x|$ is continuous.