Proving that $x^n$ converges to $0$ whenever $|x| < 1$

It suffices to show that $|x|^n=|x^n| \to 0$, so we can suppose $0\leq x <1$. Clearly the sequence is decreasing and bounded below by $0$, so it converges, say to $C$. Then $C=xC$, and since $x \neq 1$ we must have $C=0$.