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Let $$u_n := \sum_{k=1}^{n} \frac{1}{n + \sqrt{k}}.$$

Classical and easy: $u_n < 1$ and $\lim\limits_{n\to\infty} u_n = 1$.

But how to prove $u_n < u_{n+1}$ for every $n$? I have a rather long and unpleasant proof using standard calculus, but maybe some one has an elegant one?

If noone gives such a proof, I'll finally give my own.

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