Question
Let $(X, d)$ be a metric space. For each $a \in X$, define a function $f_a\colon X \to \mathbb R$ by $f_a(x) = d(x, a), (x ∈ X)$.
Prove that for all $a, b \in X$ $\sup\{|f_a(x) − f_b(x)| : x \in X\} = d(a,b).$
I know I have to prove that d(a,b) is a least upper bound but I am unsure how to go about doing it.