Let $(f_{n})_{n≥1}$ be a sequence of continuous functions verifying $$f_{n}(A)⊆A$$ for all $n≥1$. Here $A$ is a compact set. Assuming that this sequence converges uniformly to a continuous function $f$
My question is: Can we deduce that $$f(A)⊆A$$
If no, Is there some conditions on the functions $(f_{n})_{n≥1}$ in which we can get the inclusion for the limiting function $f$.
If $A$ is compact then $A$ is closed. Thus $\{f_n(x)\}$ is a sequence in $A$ that converges to $f(x)$. Since $A$ is closed, $f(x) \in A$ for all $x \in A$. Thus, $f(A) \subseteq A$.