Question

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

Answer 1

Yes.

Explanation:

Let `l = lim_(n->+oo)a_n`.
`a_n` is monotone so `b_n` will be monotone as well, and `lim_(n->+oo)b_n = lim_(n->+oo)(a_n)^2 = (lim_(n->+oo)(a_n))^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`.