# Limit of LpL^p norm

Could someone help me prove that given a finite measure space $$(X,M,σ)(X, \mathcal{M}, \sigma)$$ and a measurable function $$f:X→Rf:X\to\mathbb{R}$$ in $$L∞L^\infty$$ and some $$LqL^q$$, $$limp→∞‖\displaystyle\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$$?

I don’t know where to start.

Fix $\delta>0$ and let $S_\delta:=\{x,|f(x)|\geqslant \lVert f\rVert_\infty-\delta\}$ for $\delta<\lVert f\rVert_\infty$. We have
since $\mu(S_\delta)$ is finite and positive.
As $|f(x)|\leqslant\lVert f\rVert_\infty$ for almost every $x$, we have for $p>q$,