Inequality for Difference Between Two Harmonic Numbers

Question

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.

Answer 1

$$H_n-H_k=\sum_{j=k+1}^n\frac 1j\le\int_{k}^{n}\frac{dt}t=\log\frac nk$$