## Calculus 8th Edition

If $\lim\limits_{n\to\infty} |a_n|=0$, then $\lim\limits_{n\to\infty} a_n=0$
Prove that If $\lim\limits_{n\to\infty} |a_n|=0$, then $\lim\limits_{n\to\infty} a_n=0$ Consider $\lim\limits_{n\to\infty} |a_n|=0$ The property of absolute value is defined as: $-|a_n| \leq a_n \leq |a_n|$ for all the values of $n$. Apply Law of sequences for the limits. $\lim\limits_{n\to\infty} |a_n|=0$; $\lim\limits_{n\to\infty} -|a_n|=-\lim\limits_{n\to\infty} |a_n|=0$ According to the squeeze theorem for sequence, we have $\lim\limits_{n\to\infty} a_n=0$ Hence, it has been prove that when $\lim\limits_{n\to\infty} |a_n|=0$, then $\lim\limits_{n\to\infty} a_n=0$