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 $\sum_{n \in S} \frac{1}{n} = 1$.
[Bl21] Bloom, T. F., On a density conjecture about unit fractions.
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