If $f$: $D$-->$E$ and $g$: $E$-->$R$ are uniformly continuous, is $g º f$: $D$-->$R$ also uniformly continuous?
I think it is also uniformly continuous but I don't know how to write reasoning. can I say since the composite of two continuous functions is continuous? but it does not show that the composite is uniformly continuous.
Let $\epsilon >0$. There exists $\eta >0$ such that $d(g(u),g(v)) <\epsilon$ if $d(u,v) <\eta$. There exists $\delta >0$ such that $d(f(x),f(y)) <\eta$ if $d(x,y) <\delta$. Can you finish?