Let $(E,d)$ and $(F,d')$ be metric spaces and consider $f:E\rightarrow F$. Prove that:
$f$ is uniformly continuous in $E$ $\iff$ for every $A,B\subseteq E$ such that $d(A,B)=0$, it follows that $d'(f(A),f(B))=0$.
Where $d(A,B)=inf\{ d(x,y)|x\in A, y\in B\}$
The $\implies$ direction is easy from the definition but in the other direction I'm lost. Any help will be gratefully received.