Show that $d_{\infty}: l^{\infty} \times l^{\infty} \longrightarrow \mathbb{R}^{+}$ is well defined.

Question

I have come across some answers as to how to prove a function is well-defined. Problem is, I can't wrap my head around the fact that the definition itself here implies that the function is well-defined.

By proving this is a metric on $l^{\infty}$ wouldn't I have proved it's well defined anyway?