Let $X =\{(x_n) \subset \Bbb R \}$ such that the sequences are bounded. Define $d$: $X×X \to \Bbb R $ where $d((x_n),(y_n))=sup\{|x_n-y_n|\}$ $n \in \Bbb N$.
Prove $(X,d)$ is a metric space. The solution isn't difficult as I only need to check three trivial condition and the triangular inequality. That's not my problem. I actually wonder if $d$ is a function or not. So suprimum does exist because the sequences are bounded, right? Is that enough to say the relation $d$ is a function? Is that even necessary? Thank you very much.
Since both sequence $(x_n)_{n\in\Bbb N}$ and $(y_n)_{n\in\Bbb N}$, the sequence $(|x_n-y_n|)_{n\in\Bbb N}$ is bounded too. What this means is that the set $\{|x_n-y_n|\mid n\in\Bbb N\}$ is bounded. And, clearly, it's not empty. Since every bounded non-empty subset of $\Bbb R$ has a supremum, the definition of $d$ makes sense. In other words, yes, $d$ is a function.