Does negative infinity squared = positive infinity?

Question

I googled this question and saw this answer but I wasn't satisfied.

Answer 1

Lemma. Let $\{p_n\}_n$ be a real sequence such that $\lim_{n \to +\infty} p_n = -\infty$. Then $\lim_{n \to +\infty} p_n^2 = +\infty$.

In this sense, your conjecture is correct, and the previous lemma can be easily proved by using the mere definition of limit.

Of course, asking what the result of an algebraic operation on a symbol that is not a number is, may lead to different answers. First of all, you should ask (yourself) what is $-\infty$ and what is $+\infty$. If you are merely thinking of limits in calculus, then everything goes well. If you are dealing with infinity as a different object, you should expect some trouble from every partial extension of the usual arithmetic rules to it. But you should tell us more.