
Checking a metric space
Let $\displaystyle (X,d)$ metric space. Let $\displaystyle Y=\mathcal B\mathcal C(X,\mathbb R)$ be the space of every continuous and bounded functions from $\displaystyle X$ to $\displaystyle \mathbb R.$
Prove that $\displaystyle h(f,g)=\underset{x\in X}{\mathop{\sup }}\,\left fg \right$ is a metric on $\displaystyle Y.$
Well obviously $\displaystyle h(f,g)\ge0,$ if $\displaystyle f=g\implies h(f,g)=0,$ besides it's $\displaystyle h(f,g)=h(g,f),$ so the only thing we need to prove is that $\displaystyle h(f,g)\le h(f,j)+h(j,g).$
First, we have $\displaystyle \left fg \right=\left fj+jg \right\le \left fj \right+\left jg \right,$ but it's easy to see that every component of the inequality is bounded, so it follows that $\displaystyle \underset{x\in X}{\mathop{\sup }}\,\left fg \right\le \underset{x\in X}{\mathop{\sup }}\,\left fj \right+\underset{x\in X}{\mathop{\sup }}\,\left jg \right.$
Is this correct?