Convergence of $(x_n)$, $(x_n + y_n)$ and $(y_n)$ Prove that if real sequences $(x_n)$ and $(x_n + y_n)$ converge, then $(y_n)$ converges.  
My attempt so far:  
Suppose that the limits of $(x_n)$ and $(x_n + y_n)$ are $x$ and $x+y$ respectively. (Intuitively, $y_n \rightarrow y$ as $n \rightarrow \infty$.)
Then given any $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that $|x_n - x| < 3\epsilon/2$ and $|(x_n + y_n) - (x+y)| < \epsilon/2$ for all $n > N$. 
By triangle inequality, we have that $|(x_n + y_n) - (x+y)| \leq |x_n - x| + |y_n - y|$. I want to use this relation and $|x_n - x|$ to obtain $|y_n - y| < \epsilon$. But I haven't had any luck so far.
Any help would be appreciated! 

From what you have, you can deduce $\newcommand{\ep}{\epsilon}|y_n-y|<2\ep$. Observe that
$$|y_n-y|=|x_n+y_n-(x+y)-(x_n-x)|\le
|x_n+y_n-(x+y)|+|-(x_n-x)|<\frac32\ep+\frac\ep2.$$
By choosing $\ep/2$ instead of $3\ep/2$ you could have obtained a final
$\ep$ rather than $2\ep$.
