Limit of integral in $\mathbb R^2$

Question

I am trying to prove the following but even if it seems easy I am not sure how to start:

Let $R$ be a rectangle in the plane and $f\colon R\rightarrow {\mathbb{R}}$ a continuous non-negative function. Then the following holds:

$$\left(\iint_R f^n\right)^{\dfrac{1}{n}} \rightarrow \sup \left(f(x)\right)\text{ as }n\longrightarrow \infty.$$

Any hints will help. Thank you

Answer 1

Can you prove that the sequence $n\to (\int_{R} f^n)^{\frac{1}{n}}$ is bounded by $\sup_{x\in R} f(x)$ for all natural numbers $\mathbb{N}$?

The other direction is to show that for each $\epsilon > 0$, there exists $N$ sufficiently large such that $(\int_{R} f^n)^{\frac{1}{n}} > \sup_{x\in R} f(x) - \epsilon$ for all $n\geq N$.

Hint: Define $A_{\epsilon}=\{x\in R:f(x)>\sup_{x\in R} f(x) - \epsilon\}$. What can you say about $(\int_{A_{\epsilon}} f^n)^{\frac{1}{n}}$ for sufficiently large $n$?

Hope this helps!