# Prove that if $\lim\limits_{n \to \infty} x_n = x$ & $\lim\limits_{n \to \infty} y_n = y$ then $\lim\limits_{n \to \infty}d(x_n,y_n)$ = d(x,y)

I'm given this

Prove that if $\lim\limits_{n \to \infty} x_n = x$ & $\lim\limits_{n \to \infty} y_n = y$ then $\lim\limits_{n \to \infty}d(x_n,y_n)$ = d(x,y)

Here's what I know, d(x,y) = |x-y|

So far I've tried writing out the epsilon definitions of limits for $x_n$ and $y_n$ but I'm just not having any luck with this proof. Any pointers?

Use the inequalities that $|d(x_{n},y_{n})-d(x,y)|\leq|d(x_{n},y_{n})-d(x_{n},y)|+|d(x_{n},y)-d(x,y)|\leq d(y_{n},y)+d(x_{n},x)$.