Does the $\int_{-\infty}^\infty e^{{-x}^2} \, dx$ integral converge?

Question

I got this exercise as an optional challenge for students.

This integral is supposed to converge, and in fact, the Math Stack community has already done it, but I'm looking for something more detailed, integral there aren't shown with the detail I'd expect, hence the overall understanding of the method is somewhat difficult.

I hope I don't botter the community by asking this.

Greetings and thank you.

Answer 1

Hint $$e^{-x^2}\leq \frac{1}{x^2}.$$

Answer 2

It's often called the Gaussian integral. You can easily Google many wonderful proofs it's equal to $\sqrt{\pi}$. If all you want is a proof of convergence, use $x^2\ge |x|$ for $|x|\ge 1$, $x^2\ge 0$ for all $x$.