In general, one may not interchange limits (eg, $\lim_k f_k(\lim_n x_n)\neq \lim_n\lim_k f_k(x_n)$ in most cases) without requiring that the family $\{f_k\}$ is uniformly continuous. However, can you help me in the following situation?
Let $X$ be a compact metric space (think unit interval $[0,1]$ if you prefer), with $f:X\rightarrow X$ a homeomorphism (so, 1-1 continuous with continuous inverse). We do not know that the family $\{f^n\}_{n\in\mathbb{Z}}$ is uniformly continuous, but we do know that there is a convergent sequence $\{f^{k'}\}$ (where by convergent we mean pointwise convergent) and that there is also a convergent sequence $\{f^{n'}(x)\}$ (where here convergence is of a sequence of points in $X$ with its compact metric topology). Are we allowed to do the following: $$ \lim_{k'\rightarrow +\infty} f^{k'}(\lim_{n'\rightarrow+\infty}f^{n'}(x))=\lim_{n'\rightarrow+\infty}\lim_{k'\rightarrow+\infty} f^{k'+n'}(x)= \lim_{n'\rightarrow +\infty} f^{n'}(\lim_{k'\rightarrow+\infty}f^{k'}(x)) $$
I think the answer is 'no', but a counterexmple would be useful. I've also looked at this question, but it's not quite what I'm looking for. Thanks!