sequences and series - can I say that $lim(x_n-y_n) = x - y$ if $x=lim(x_n)$, $y=lim(y_n)$?

I have this question in my HW: true or false,

If $x_n$ is any increasing sequence of negative real numbers and $y_n$ is a cauchy sequence of real numbers, then the sequence $x_n-y_n$ converges.

My guess it's true. for example $x_n=\{-1/n\}$ which is an example of increasing sequence of real number. In this case $x_n$ converges to $x=0$ and since $y_n$ is cauchy so it converges to a real number $y$, but how can I prove this. I know that $\lim(x_n+y_n)=x+y$ can I use it as $\lim(x_n-y_n)=x-y$?