# Suppose, a_n is monotone and converges and b_n=(a_n)^2. Does b_n necessarily converge?

Let $l = {\lim}_{n \to + \infty} {a}_{n}$.
${a}_{n}$ is monotone so ${b}_{n}$ will be monotone as well, and ${\lim}_{n \to + \infty} {b}_{n} = {\lim}_{n \to + \infty} {\left({a}_{n}\right)}^{2} = {\left({\lim}_{n \to + \infty} \left({a}_{n}\right)\right)}^{2} = {l}^{2}$.
It's like with functions : if $f$ and $g$ have a finite limit at $a$, then the product $f . g$ will have a limit at $a$.