Answer
$\lim\limits_{n\to\infty} (a_nb_n)=0$
Work Step by Step
Need to prove that $\lim\limits_{n\to\infty} (a_nb_n)=0$
When a composition of a function is continuous , then the limit also becomes continuous. Let us consider that $\lim\limits_{n\to\infty} a_n=L$ and the function $f(a_n)$ is continuous at the limit $L$.
The property of absolute value: $-|a_n| \leq a_n \leq |a_n|$ for all the values of $n$, thus $\lim\limits_{n\to\infty} |a_n|=0$
According to the limits laws of sequences:
$\lim\limits_{n\to\infty} |a_n|=0$; $\lim\limits_{n\to\infty} -|a_n|=-\lim\limits_{n\to\infty} |a_n|=0$
Apply the squeeze theorem for sequence.
$\lim\limits_{n\to\infty} a_n=0$. Thus, If $\lim\limits_{n\to\infty} |a_n|=0$, then $\lim\limits_{n\to\infty} a_n=0$
Need to consider a small number $\epsilon \gt 0$ , then $a_n \to 0$
This means that there exists a number $N$ such that the value $|a_n| \lt \dfrac{\epsilon }{M}$ for the all values of $n \geq N$
This gives: $|a_nb_n|=|a_n||b_n| \lt \dfrac{\epsilon }{M} (M)=\epsilon$
So, this is verified that $\lim\limits_{n\to\infty} (a_nb_n)=0$
