IMO 1978 — Official contest problems Question 5

Let $\varphi:\{1,2,3,\ldots\}\to\{1,2,3,\ldots\}$ be injective. Prove that for all $n$,
$$\sum_{k=1}^n\frac{\varphi(k)}{k^2}\ge\sum_{k=1}^n\frac1k.$$