Family not satisfying uniform equicontinuity

Question

Suppose that a family of homeomorphisms $f_n$ from the compact metric space $(X,d)$ to the compact metric space $(Y,d')$ is not uniformly equicontinuous. Does this imply that there exists two different points $x,y$ and a subsequence $n_j$ such that $d'(f_{n_j}(x),f_{n_{j}}(y))\rightarrow 0$ as $j \rightarrow \infty$?