I want to know if the following inequality holds: $H_n-H_k \leq \log{n}-\log{k}$ where $k \leq n$ and $n \to \infty$.
I have seen two similar questions such as this question and this one but I could not verify if my statement is correct.
2026-03-29 14:27:16.1774794436
Inequality for Difference Between Two Harmonic Numbers
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$$H_n-H_k=\sum_{j=k+1}^n\frac 1j\le\int_{k}^{n}\frac{dt}t=\log\frac nk$$