Answer
If $\lim\limits_{n\to\infty} |a_n|=0$, then $\lim\limits_{n\to\infty} a_n=0$
Work Step by Step
We need to prove that if $\lim\limits_{n\to\infty} |a_n|=0$, then $\lim\limits_{n\to\infty} a_n=0$
Consider the property of the absolute value as follows:
$-|a_n| \leq a_n \leq |a_n|$ for all the values of $n$
Let us consider that $\lim\limits_{n\to\infty} |a_n|=0$
The limit laws of sequences state that
$\lim\limits_{n\to\infty} |a_n|=0$; ( for all the values of $n$, we have $-|a_n| \leq a_n \leq |a_n|$ )
and $\lim\limits_{n\to\infty} -|a_n|=-\lim\limits_{n\to\infty} |a_n|=0$
The squeeze theorem for a sequence states that $\lim\limits_{n\to\infty} a_n=0$
Hence, it has been verified that when $\lim\limits_{n\to\infty} |a_n|=0$, then $\lim\limits_{n\to\infty} a_n=0$
