Proof request of $p(A) < 1 \space \text{iff} \space \lim_{k \to \infty} A^k = 0$

Question

I'm seeking a proof to the following lemma that I found being stated without being proven at my Numerical Linear Algebra course textbook, which involves a correlation between the spectral radius of a matrix $A$, $p(A)$ and the limit of the power of $A$ :

Let $p(A)$ denote the spectral radius of the matrix $A$.

Then, it is $p(A) <1$ if and only if $\lim_{k \to \infty} A^k = 0$.