## Checking a metric space

Let $(X,d)$ metric space. Let $Y=\mathcal B\mathcal C(X,\mathbb R)$ be the space of every continuous and bounded functions from $X$ to $\mathbb R.$

Prove that $h(f,g)=\underset{x\in X}{\mathop{\sup }}\,\left| f-g \right|$ is a metric on $Y.$

Well obviously $h(f,g)\ge0,$ if $f=g\implies h(f,g)=0,$ besides it's $h(f,g)=h(g,f),$ so the only thing we need to prove is that $h(f,g)\le h(f,j)+h(j,g).$

First, we have $\left| f-g \right|=\left| f-j+j-g \right|\le \left| f-j \right|+\left| j-g \right|,$ but it's easy to see that every component of the inequality is bounded, so it follows that $\underset{x\in X}{\mathop{\sup }}\,\left| f-g \right|\le \underset{x\in X}{\mathop{\sup }}\,\left| f-j \right|+\underset{x\in X}{\mathop{\sup }}\,\left| j-g \right|.$

Is this correct?