Is there an infinite sequence $a_1 < a_2 < \dots$ such that $a_{i+1} - a_i = O(1)$ and no finite sum of $\frac{1}{a_i}$ is equal to 1?

The corresponding question is also false if one replaces sequences such that $a_{i+1} - a_i = O(1)$ with sets of positive density, as follows from [Bl21].

The statement is as follows: If $A \subset \mathbb{N}$ has positive upper density (and hence certainly if $A$ has positive density) then there is a finite $S \subset A$ such that $\sum_{n \in S} \frac{1}{n} = 1$.