Solution from Peter Linz itself:

For $n = 1,$ $|u^1| = |u|.$ Assume $|u^k| = k|u|$ holds for $k = 1, 2, ..., n,$ then

$|u^{n+1}| = |u^nu| = |u^n| + |u| = n|u| + |u| = (n + 1)|u|.$

Detailed Video Solution