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Statement and provenance

If $a_1<a_2<\cdots$ are positive integers and $a_{i+1}-a_i=O(1)$ as $i\to\infty$, then some finite set of indices $I$ satisfies $\sum_{i\in I}1/a_i=1$.

This is the consequence of Bloom's Theorem 2 identified by the cached Problem 299 page, accessed 2026-09-05. It is not a separately numbered result in Bloom's paper. It disproves the existence asked for in Problem 299.

Rewritten derivation

Choose integers $i_0\ge1$ and $H\ge1$ so that $a_{i+1}-a_i\le H$ for every $i\ge i_0$. Induction gives $a_{i_0+j}\le a_{i_0}+Hj$ for all $j\ge0$. Consequently the set $A=\{a_i:i\ge1\}$ satisfies, for $N\ge a_{i_0}$,

$$|A\cap[1,N]| \ge1+\left\lfloor\frac{N-a_{i_0}}H\right\rfloor.$$

Indeed, all indices $i_0+j$ through the indicated floor have $a_{i_0+j}\le N$, and strict increase makes their values distinct. Divide by $N$ and take the lower limit to obtain $\underline d(A)\ge1/H>0$. Hence $\overline d(A)>0$ as well. Theorem 2 supplies a finite $S\subseteq A$ of reciprocal sum one. Each element of $S$ has a unique index because the sequence is strictly increasing; taking those indices gives the required $I$.

Existing formalization

The Google DeepMind declaration for Problem 299 uses a strictly increasing positive sequence and an eventual big-O gap bound. As inspected on 2026-09-05, its body is sorry, with an external formal-proof tag linking the Bloom–Mehta Lean 3 density proof. That link supports the density theorem; no separate completed Lean declaration for the bounded-gap reduction was identified in the inspected source. No formal proof was built here.

Dependencies

Theorem 2 only; its full proof is kept on its own page and is not repeated here.

Bears on

• Problem 298
• Problem 299