Can I write $\sup_{i \in I} \liminf_{n} f_{i}(x_{n}) \leq \liminf_{n} f_{i} \sup_{i \in I} (x_{n})$ if $f_{i}$ is a continuous function ?
2026-03-28 02:22:08.1774664528
Interchange the lim inf and supremum
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Yes (assuming I'm interpreting your desired formula correctly). For any fixed $j$ and any fixed $n$, it's obvious that $$ f_j(x_n) \le \sup_i f_i(x_n). $$ Taking lim infs of both sides yields $$ \liminf_n f_j(x_n) \le \liminf_n \sup_i f_i(x_n); $$ note that the right-hand side is now a constant, independent of the fixed $j$. Taking sups of both sides (doesn't affect the right-hand side) and yields $$ \sup_j \liminf_n f_j(x_n) \le \liminf_n \sup_i f_i(x_n), $$ which is presumably what you are wanting to prove (and changing $j$ to $i$ on the left-hand side is fine because it's a dummy variable).