Let $X$ a metric space, $K_{n}\subset X$ compact and
$f_{n}:K_{n}\longrightarrow K_{n}$ a continuous function, for each $n\geq 1$.
It is easy to prove that if $f_{n}\rightarrow f$ uniformly, then $d_{H}(X,K_{n})\rightarrow 0$, $d_{H}$ being the Hausdorff distance. But, Under that conditions we can state the reciprocal? That is, if $d_{H}(X,K_{n})\rightarrow 0$, Under that conditions can we ensure the existence of $f:X\longrightarrow X$ such that $f_{n}\rightarrow f$?
Many thanks in advance for your comments.