<s>[INST] Show That the Series $\frac{1}{n}+ \frac{1}{n+1} +\dots + \frac{1}{2n}$ Converges to a Number Between $\frac{1}{2}$ and $1$ $a_n = \frac{1}{n}+ \frac{1}{n+1} +\dots + \frac{1}{2n}$ converges to a number between $\frac{1}{2}$ and $1$.
I know that it converges to Ln(2) but we still haven't learned about integrals in my class,  so calculating its limits kills the purpose of the exercise.
I've been trying for a while, I tried with this theorem that states that if $a_n \leq b_n$ for all n greater or equal to some N, then $\lim_{n\to\infty} a_n \leq \lim_{n\to\infty} b_n$
So my aproach was to find two sequences such that their limits are $\frac{1}{2}$ and $1$, I used these $\frac{n}{2n+1}$ and  $\frac{n}{n+1}$, the limit of the first one is $\frac{1}{2}$ and the limit of the second one  is $1$, and we have that 
$$\frac{n}{2n+1} \leq \frac{1}{n}+ \frac{1}{n+1} +\dots + \frac{1}{2n} \leq \frac{n}{n+1}$$ for all $n \geq 6$, I've been trying to prove that with induction but I have been unable to do it. Is there any other way to prove that the limit is between $\frac{1}{2}$ and $1$?, or if someone could give me a hint with the induction proof, I would be very grateful.
 [/INST] Hint 1:
$$a_{n+1}-a_n=\left( \frac{1}{n+1}+ \frac{1}{n+1} +\dots + \frac{1}{2n+2} \right)-\left(\frac{1}{n}+ \frac{1}{n+1} +\dots + \frac{1}{2n}\right)\\ =  \frac{1}{2n+1}+\frac{1}{2n+2}- \frac{1}{n}<\frac{1}{2n}+\frac{1}{2n}- \frac{1}{n}=0$$
Hint 2:
$$a_n =\frac{1}{n}+ \frac{1}{n+1} +\dots + \frac{1}{2n} \geq \frac{1}{2n}+ \frac{1}{2n} +\dots + \frac{1}{2n}$$
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