  We prove that the existence of infinitely many $(m_k, n_k) \in \mathbb{N}^2$
such that the difference of harmonic numbers $H_{m_k} - H_{n_k}$ approximates 1
well $$ \lim_{k \rightarrow \infty} \left| \sum_{\ell = n}^{m_k} \frac{1}{\ell}
- 1 \right|\cdot n_k^2 = 0.$$ This answers a question of Erd\H{o}s and Graham.
The construction uses asymptotics for harmonic numbers, the precise nature of
the continued fraction expansion of $e$ and a suitable rescaling of a
subsequence of convergents. We also prove a quantitative rate by appealing to
techniques of Heilbronn, Danicic, Harman, Hooley and others regarding $\min_{1
\leq n \leq N} \min_{m \in \mathbb{N}}\| n^2 \theta - m\|$.
