I have an assignment to precisely describe epimorphisms and monomorphisms in Met (category whose objects are Metric spaces and whose morphisms are contractions).
I have shown that Mono $\iff$ one-to-one, and also that "having dense image" $\implies$ epi.
I know that the converse of the latter is true, but I have been unable to prove it. For the direction I did prove I used a convergent sequence and continuity of the morphisms (given they are contractions). However, I do not know how to use the fact that a morphism can be cancelled on the right to show that it has dense image. Any pointers are appreciated.
Suppose $f:X\to Y$ is such that $f(X)$ is not dense. That is, let $B(x,r)$ be an open ball in $Y$ which $f(X)$ does not intersect. Let $g_1:Y\to \mathbb{R}$ be $g_1(y)=0$, and $g_2:Y\to \mathbb{R}$ be $g_2(y)=\max(r-d(y,x),0)$. The maximum of continuous functions is continuous, so the $g_i$ are continuous functions which agree on $f(X)$ but don't agree on $Y$ and $f$ is not an epimorphism.