Q:

Proof of limit in analysis

I am trying to prove that if $|a| < 1$, then, $a^n \to 0$ as $n \to \infty$. The hint says that it suffices to prove that $|a|^n \to 0$ as $n \to \infty$. Why is that?

A:

In general, a sequence $\{x_n\}$ converges to $l$ if and only if $\{|x_n - l|\}$ converges to 0. (That fact follows easily from the definition of convergence.) In your case, we have:
\begin{align*}
  a^n \to 0
  &\iff |a^n - 0| \to 0 \\
  &\iff |a^n| \to 0 \\
  &\iff |a|^n \to 0.
\end{align*}

