<s>[INST] Prob. 10, Sec. 3.3, in Bartle & Sherbert's INTRO TO REAL ANALYSIS, 4th ed: Is this sequence convergent? Here is Prob. 10, Sec. 3.3, in the book Introduction to Real Analysis by Robert G. Bartle & Donald R. Sherbert, 4th edition: 

Establish the convergence or the divergence of the sequence $\left( y_n \right)$, where 
  $$ y_n \colon= \frac{1}{ n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n} \ \mbox{ for } n \in \mathbb{N}. $$

My Attempt: 

For each $n \in \mathbb{N}$, we have 
  $$
\begin{align}
y_{n+1} - y_n &= \left( \frac{1}{ (n + 1) +1} + \frac{1}{ (n+1) +2 } + \cdots + \frac{1}{2(n+1)} \right) - \left( \frac{1}{ n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n} \right) \\
&= \left( \frac{1}{ n  + 2} + \cdots + \frac{1}{2n+ 2 } \right) - \left( \frac{1}{ n+1} + \cdots + \frac{1}{2n} \right) \\
&= \frac{ 1 }{2n+1} + \frac{ 1}{2n+2} - \frac{1}{n+1} \\
&= \frac{1}{2n+1} - \frac{1}{2n+2} \\
&= \frac{1}{ (2n+1) (2n+2) } > 0,
\end{align}
$$
  which shows that our sequence is monotonically increasing. 
Also, for each $n \in \mathbb{N}$, we have
  $$ y_n = \frac{1}{ n+1} + \frac{1}{n+2} + \cdots + \frac{1}{2n} < \underbrace{\frac{1}{n} + \frac{1}{n} + \cdots + \frac{1}{n} }_{ \mbox{ $n$ times } }  = 1. $$
Thus our sequence, being monotonically increasing and bounded, is convergent. 

Is what I've done so far all correct? 
How to determine the limit of this sequence? 
 [/INST] Your proof is correct. 
It is too early (in the book) to obtain the limit. Some time later it will be easy to see an integral sum of
$$
\int_{0}^{1}\frac{1}{1+x}\,dx.
$$
</s>