So, I am working on problems on $L_p$ spaces. Every time I think of taking limits of both sides of an equation, the solution seems to take limit inferior/ limit superior instead and use the relation between them to prove an equality.
As an example, I had for all $p$ and arbitrary $M\leq\|f\|_\infty$,
$$\|f\|_p \geq \mu(E)^{1/p}M$$
where the norm is the $L_p$ norm of a function on $(X,\mu)$ and $E$ is a set of finite measure.
Now I was going to take the limit of both sides with $p\to \infty$, and then take limit of $M$ going to $\|f\|_\infty$ to show that $$\lim_{p\to\infty}\|f\|_p\geq \|f\|_\infty$$
but the solutions specifically take $\liminf_{p\to\infty}$.
I am confused on why they do this? Am I making a fundamental mistake?
Thanks!
Answer
We take the $\liminf$ because we don't know a priori the limit exists, while the $\liminf$ (possibly infinity) always exists.
(here it turns out that the limit indeed exists, we we don't know it yet at this step)
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