Element of a group G raise to the index of a subgroup H is in H

It's only a superficial change, but since you seem to be having issues accepting your (correct) argument, maybe you will like this better:

Instead of talking (explicitly) about cosets, let's talk (equivalently, of course!) about the quotient group and the quotient map $q: G \rightarrow G/H$. For $x \in G$, since $\# (G/H) = m$, by Lagrange's Theorem, $q(x)^m = e$ (the identity) in $G/H$. But since $q$ is a homomorphism, $e = q(x)^m = q(x^m)$, so $x^m \in H$.

In any case: believe it! It's an extremely useful result: I have used it many times in my work in algebra and number theory and have several times noted that students don't seem to be as aware of it as they should be. (In fact, I like to think of this in terms of the period-index problem in Galois cohomology, but that's really beyond the scope of this answer.)