How do I show that $a_{n}= \frac{1}{n + 1} + \frac{1}{n + 2} + ... + \frac{1}{2n}$ for n ≥ 1, is bounded by using $a_{n} ≤ \frac{n}{n+1}$ where n ≥ 1.
I know to show that a sequence is bounded you show it is bounded both above and below but I am unsure as to how to use $a_{n} ≤ \frac{n}{n+1}$ where n ≥ 1
Any help would be appreciated
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If you know harmonic numbers and their properties $$a_n=\sum_{k=1}^n\frac 1{n+k}=H_{2n}-H_n$$ Now, using the asymptotics $$H_p=\gamma +\log \left({p}\right)+\frac{1}{2 p}-\frac{1}{12 p^2}+O\left(\frac{1}{p^3}\right)$$ apply it twice to get $$a_n=\log (2)-\frac{1}{4 n}+\frac{1}{16 n^2}+O\left(\frac{1}{n^3}\right)$$ In fact, it is an alternating series and then $$\log (2)-\frac{1}{4 n} < a_n <\log (2)-\frac{1}{4 n}+\frac{1}{16 n^2}$$
$0\lt a_n \le \dfrac{n}{n+1} \lt 1$