# Exchanging limsup with liminf

I am working on a problem where I think I might be able to complete my argument if I can show the following relation.

\begin{align*} \limsup_{k \rightarrow \infty} \liminf_{x' \rightarrow x} \frac{1}{k} C(k, x') &\le \liminf_{x' \rightarrow x} \limsup_{k \rightarrow \infty} \frac{1}{k} C(k, x'). \end{align*}

Here $$k$$ is an integer approaching infinity. (The function $$C(k, x')$$ is defined on integers $$k$$, not real numbers). I have not encountered such a double-limit problem before and could not find a good resource either. Would it be possible to show this relation under certain conditions. If it helps in anyway, for every fixed $$k$$, the function $$C(k, x')$$ is lower semi-continuous in $$x'$$ and bounded from below. Also, the space of $$x'$$ may be assumed to be a complete separable metric space.