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? 

What you did is fine.
Now, as far as the imit is concerned, note that$$y_n=\frac1n\left(\frac1{1+1/n}+\frac1{1+2/n}+\cdots+\frac1{1+n/n}\right).$$This is a Riemann sum.
