I have the following problem:
Let S be the set of bounded functions on [a,b] with $d(x,y) = Sup_{a\leq t \leq b} |x(t)-y(t)|$. Show that S is a metric space.
I think that non-negativity and symmetry both follow immediately from the properties of absolute value here, yes?
Further, can I verify $d(x,y)\leq d(x,z) + d(z,y)$ in the following way:
$$d(x,y)=Sup_{a \leq t \leq b}|x(t)-y(t)| = Sup_{a \leq t \leq b}|x(t)- z(t) + z(t) - y(t)| $$$$\leq Sup_{a \leq t \leq b}(|x(t)-z(t)|+|z(t)-y(t)|)$$ $$\leq Sup_{a \leq t \leq b}(|x(t)-z(t)|)+Sup_{a \leq t \leq b}(|z(t)-y(t)|)$$ $$ = d(x,z)+d(z,y)$$
Is this the correct approach?
