Going from max of set to l.u.b. of a set in proving a metric

Question

We have to verify the metric $d(f,g)=l.u.b\:\cup_{x\in [a,b]}\{|f(x)-g(x)|\}$ for f,g continuous satisfies the Schwartz inequality (triangle inequality). It is easy to prove that if F and G are functions of a discrete variable $d(F,G)=max\: \cup_x\{|f(x)-g(x)|\}$ satisfies the inequality, but I stumble on the corresponding proof for functions of a continuous variable using least upper bound.