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.



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