From Fitzpatrick's Advanced Calculus book: "Suppose that the sequence ${\{a_n}\}$ is monotone, i.e., either monotonically increasing or decreasing. Prove that ${\{a_n}\}$ converges if and only if ${\{{a_n}^2}\}$ converges. Show that this result does not hold without the monotonicity assumption."
It is easy to show to that ${\{{a_n}^2}\}$ converges if ${\{a_n}\}$ converges: ${\{a_n}\}$ is convergent sequence and ${\{a_n}\}$ also, since product of two convergent sequences converges (to the product of their limit), so ${\{{a_n}^2}\}={\{a_n\times a_n}\}$ converges; and it doesn't require that ${\{a_n}\}$ to be monotone.
On the other hand, it is not general true to say that ${\{a_n}\}$ converges if ${\{{a_n}}^2\}$ converges; ${\{a_n}\}={\{(-1)^n}\}$ is a counterexample.
What remains to evaluate (i.e., my questions) is that:
$1-$ Prove that if ${\{{a_n}^2}\}$ converges and if ${\{a_n}\}$ is monotone, thus ${\{a_n}\}$ converges.
And,
$2-$ If ${\{{a_n}^2}\}$ converges to $a^2$, to which of $a$ or $-a$ does ${\{a_n}\}$ converge?
Thank you.
Best Answer
If $\{a_n^2\}$ converges, it is bounded. Then $\{a_n\}$ is bounded and monotone.
Anything can happen. There is no way to know a priori.