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I am trying to find a number sequence $x_k\geq 0$ such that $$\text{for all integers }n\geq k\geq 1,\quad H_{n-1 - x_k(n-k)}\geq H_n-\frac{1}{k}.$$ I recall that the harmonic numbers $s\mapsto H_s$ can be defined for $s>-1$ by $H_s=\int_0^1\frac{1-t^s}{1-t}dt$. $H:(-1,\infty)\to\mathbb{R}$ is a bijection.

In particular, if $H^{-1}: \mathbb{R}\to (-1,\infty)$ is the inverse function of $H$, how can I prove that $x_k=-H^{-1}_{-1/k}$ works ?

Thank you!

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