Convergent + divergent $\to$ divergent Given sequences $(x_n)$, convergent, but $(y_n)$ is divergent, then $(x_n + y_n)$ is divergent.
I am confident that it is true, but having trouble getting the formalities correct. I have tried proof by contradiction, i.e. assuming 
$\forall \varepsilon > 0 \; \exists N \in \mathbf{N}:$
$$ |(x_n + y_n) - L| < \varepsilon \qquad  \forall n > N.$$
This seems to lead me nowhere. Equivalently, trying to find an $\varepsilon$ such that for all $N$ $|(x_n + y_n) - L| \geq \varepsilon$ have not gotten me any further.  Any hints are appreciated.

If $(a_n)$ and $(b_n)$ are convergent, then so is $(a_n+b_n)$ and $(ca_n)$ for any scalar $c$. 
Now, assume that $(x_n)$ and $(x_n+y_n)$ are convergent to deduce that $(x_n+y_n-x_n)=(y_n)$ is convergent.
