<s>[INST] Prove that $\frac 12\leq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}...+\frac{1}{n+n}$ How do I prove $$\frac 12\leq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}...+\frac{1}{n+n}$$ 
without using induction? Note that clearly $n\neq 0$
Thanks for any help!!
 [/INST] This inequality does not hold since for any $n\in \mathbb N$ you have 
$$\frac{1}{2}\leq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}+...+\frac{1}{n+n}.$$
To see this, note first that any term above is greater than $\frac{1}{n+n}$, so that your sum is greater than $n\cdot \frac{1}{2n}=\frac{1}{2}.$ 
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