I came across an International Mathematical Olympiad problem from 1978 which I had the feeling I would be able to solve quickly. The problem is stated as follows:

Problem (IMO ‘78)

Suppose that \(n\) is a positive integer, and that \( (a _ k) \) is a one-one sequence of positive integers. Then \[ \sum _ {k=1} ^{n} \frac{1}{k} \leq \sum _ {k=1}^{n} \frac{a _ k} {k^2} \].

In order to prove this fact I will use two elementary inequalities.

Lemma (Cauchy-Schwarz inequality)

Suppose that \( (a _ k), (b _ k) \) are sequences of real numbers. Then for any positive integer \( n\) we have

\[ \sum _ {k=1} ^ {n} a _ k b _ k \leq \sqrt{\sum _ {k=1} ^ {n} a _ k ^2 } \sqrt{\sum _ {k=1} ^{n} b _ k ^2 } \]

completing the proof. Note that the last inequality is where I used the fact that the sequence \( (a _ k) \) was a one-one sequence of positive integers.