CMO 1973

For every positive integer $n$ , let $$h(n) = 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}.$$ For example, $h(1) = 1$ , $h(2) = 1+\frac{1}{2}$ , $h(3) = 1+\frac{1}{2}+\frac{1}{3}$ . Prove that for $n=2,3,4,\ldots$ $$n+h(1)+h(2)+h(3)+\cdots+h(n-1) = nh(n).$$