SIMON-MARAIS 2020 B2

For each positive integer $k$, let $S_k$ be the set of real numbers that can be expressed in the form

$$\frac{1}{n_1} + \frac{1}{n_2} + \cdots + \frac{1}{n_k},$$

where $n_1, n_2, \ldots, n_k$ are positive integers.

Prove that $S_k$ does not contain an infinite strictly increasing sequence.