<s>[INST] Prove inequality $1 < \frac{1}{n} + \frac{1}{n+1} + \ldots + \frac{1}{3n-1} < 2$ Prove the inequality $1 < \frac{1}{n} + \frac{1}{n+1} + \ldots + \frac{1}{3n-1} < 2$ For all $n \in \mathbb{N}$
I've done the right hand side, but can't do the left side of the inequality. For the right:
$\frac{1}{n} + \ldots + \frac{1}{3n-1} < \underbrace{\frac{1}{n} + \ldots + \frac{1}{n}}_{2n} = 2$
Now I haven't been able to make progress with the left.
 [/INST] Hint:
Show $\dfrac{1}{n} \lt \dfrac{1}{n+k}+\dfrac{1}{3n-1-k} \lt \dfrac{2}{n}  $ for $0 \le  k \lt n$
Then sum over $k$
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