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Let $(x_n)$ be a sequence of real numbers.

I am wondering we can formally prove that:

$ \lim_{n\rightarrow\infty}|x_n| < a$

for all $a > 0$, implies

$ \lim_{n\rightarrow\infty}|x_n| = 0$

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    robjohn's comment is$a$rephrasing of (a very slightly weaker version of) Lopsy's comment.2012-12-14