Show that $S:c_0\to c_0$ given by $S( x_1, x_2,....) = (0,x_1,x_2,..)$ is an isometry for the $l_\infty$ norm

Let $S:c_{0}\to c_{0}$ be given by $S( x_1, x_2,....) = (0,x_1,x_2,..)$. Show that $S$ is an isometry,where $c_0$ is the subspace of $l_{\infty}$ consisting of sequences which converge to $0$.

My attempt: to prove isometry I need to prove that for any $x,y \in c_0$, $l_{\infty}{(x,y)} = l_{\infty}{(S(x), S(y))}$.

But $l_{\infty}{(x,y)} = \sup\{ | x_{i} - y_{i} | ,\,i \in \mathbb{N} \}$

Thus $l_{\infty}{(x,y)} = l_{\infty}{(S(x), S(y))}$.

But I don't understand where I used that $x$ and $y$ converges to $0$, I think we don't need that.