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! 

Set $L$ to be the limit of $(x_{n}+y_{n})$ (or we take $y=L-x$ as OP noted), and we claim that $y=L-x$ is the limit of $(y_{n})$: For $\epsilon>0$, choose $N$ such that for all $n\geq N$, $|x_{n}-x|<\epsilon/2$ and $|x_{n}+y_{n}-L|<\epsilon/2$, then for such an $n$, we have 
\begin{align*}
|y_{n}-(L-x)|&=|y_{n}+x_{n}-L-x_{n}+x|\\
&\leq|x_{n}+y_{n}-L|+|x_{n}-x|\\
&<\epsilon.
\end{align*}
