I'm trying to calculate with nested $\inf$ and $\sup$, but I feel like I don't know the right tricks and missing the right tools of calculation with this. I wish there are rules of transfromation like in set theory. Does anyone how to calulate with nested $\inf$ and $\sup$ cleverly? I konw the common propertys like monotonicity, but it doesn't help me in my case.
The problem I'm dealing with is the following: Let $f:[0,\infty[\to[0,\infty[$ a continous funtion. I'm asking myself if $$\inf\limits_{k\in\mathbb{N}_0}\ \sup\limits_{m\geq k}\ \inf\limits_{t\in [m,m+1[} f(t)=\sup\limits_{a>0}\ \inf\limits_{t\in ]a,\infty[} f(t) $$ holds true and how to prove or disproof it. I know that the statement is aquivalent to $$\limsup\limits_{m\to \infty} \inf\limits_{t\in [m,m+1[} f(t) = \liminf\limits_{t\to\infty} f(t)$$ For any kind of help I'm very grateful.