TITLE: $\sup_{m>0}( \liminf_{n \to \infty} f_{m,n}) \le \liminf_{n \to \infty} (\sup_{m>0}f_{m,n}) ?$
QUESTION [0 upvotes]: $$\sup_{m>0}( \liminf_{n \to \infty} f_{m,n}) \le \liminf_{n \to \infty} (\sup_{m>0}f_{m,n}) $$ 
$$f_{m,n}:R \to [0, \infty)$$
When is the inequality true ?

REPLY [3 votes]: $\forall m,f_{m,n}\leq \sup_{m>0}f_{m,n} \rightarrow \forall m,\varliminf\limits_{n\to \infty}f_{m,n} \leq\varliminf\limits_{n\to \infty}\sup_{m>0}f_{m,n} \rightarrow \sup_{m>0}(\varliminf\limits_{n\to \infty}f_{m,n}) \leq\varliminf\limits_{n\to \infty}\sup_{m>0}f_{m,n}$ 
So I think it's true.
{"set_name": "stack_exchange", "score": 0, "question_id": 2873923}