<s>[INST] How do I prove $\frac 34\geq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{n+n}$ How do I prove the following inequality
$$\frac 34\geq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{n+n}$$
without the help of induction?
Thanks for any help!!
 [/INST] $f(x)=\frac{1}{n+x}$ is a convex function on $\mathbb{R}^+$, hence:
$$ \frac{f(1)+f(2)+\ldots+f(n)}{n}\geq f\left(\frac{1+2+\ldots+n}{n}\right) \tag{1} $$
implies:
$$ \frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+n}\geq \frac{n}{n+\frac{1+2+\ldots+n}{n}}=\frac{2n}{3n-1}.\tag{2}$$
On the other hand, the Hermite-Hadamard inequality gives:
$$ \frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+n}\leq \int_{n+\frac{1}{2}}^{2n+\frac{1}{2}}\frac{dx}{x} = \log\left(\frac{4n+1}{2n+1}\right)\leq \log 2.\tag{3}$$
An alternative approach is to prove first that the sequence $\{a_n\}_{n\geq 1}$ given by $a_n=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{2n}$ is increasing, then to compute its limit at $n\to +\infty$ 
and check it is less than $\frac{3}{4}$.
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