Let X be a set, How to show $\lim\limits_{x \to \infty} d(x_n,y_n) = d(x,y)$ for two sequences $x_n,y_n \in X$ and $ x,y \in X$ Let $(X, d)$ be a metric space, let $x_n$ and $y_n$ be two sequences in $X$ and $x, y ∈ X$. Suppose that
$x_n$ converges with respect $d$ to $x$ and $y_n$ converges with respect to $d$ to $y$. Show that $\lim\limits_{x \to \infty} d(x_n,y_n) = d(x,y)$
hint : use triangle inequality several times.
I attempted this question by applying triangle inequality to the $d(x,y)$, where I get for $x,y,z \in X$, $d(x,y) \leq d(x,z) + d(z,y)$. but I don't see this leading anywhere.
Intuitively, I also feel like there is no need to apply the triangular inequality. Isn't it true that since $x_n \to x, y_n \to y$, I can write 
$\lim\limits_{x \to \infty} d(x_n,y_n) = d(x,y)$ and be done with it?
Thank you for the help guys.

$$d(x_n,y_n)\le d(x_n,x)+d(x,y)+d(y,y_n)$$ 
and 
$$d(x,y)\le d(x,x_n)+d(x_n,y_n)+d(y_n,y)$$ 
Now apply the limits.
