<s>[INST] Need help with an inequality $1<\frac{1}{1001}+\frac{1}{1002}+...+\frac{1}{3001}<1\frac{1}{3}$
The first part is trivial with $AM-HM $ inequality. Having problem with the second part. 
 [/INST] For $0 \leqslant k \leqslant 1000$ 
$$\frac{1}{2001-k} + \frac{1}{2001+k} = \frac{4002}{2001^2 - k^2} \leqslant \frac{4002}{2001^2 - 1000^2} = \frac{4002}{1001 \cdot 3001} < \frac{1}{1000 \frac{1}{2}} \cdot \frac{4}{3}.$$
Adding up the inequalities for $k = 1, 2, \ldots, 1000$ and the inequality divided by $2$ for $k = 0$, we obtain
$$\frac{1}{1001} + \frac{1}{1002} + \ldots + \frac{1}{3001} < \frac{1000 \frac{1}{2}}{1000 \frac{1}{2}} \cdot \frac{4}{3} = \frac{4}{3}$$
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