Proving Limit of Integral for Nonnegative Continuous Function on [0,1]

[adpc]

Homework Statement

f nonnegative continuous function on the interval [0,1]. Let M be the supremum of f on the interval. Prove:
$$\lim_{n \rightarrow \infty} \left[ \int^1_0 f(t)^n dt \right] ^{1/n} = M$$

Homework Equations

The Attempt at a Solution

I was trying using Upper Sums but I don't know how to compute the limit.

 

[micromass]

It seems like you need following intermediate result:

Let $$x_1,...x_n$$ be real numbers and $$w_1,...,w_n$$ be positive such that $$w_1+...+w_n=1$$. Assume without loss of generality that $$x_1$$ is the greatest among the $$x_i$$. Then

$$\lim_{n\rightarrow+\infty}{\sqrt[n]{\sum_{i=1}^k{w_ix_i^n}}}=x_1$$.

HINT: Show, using the squeeze theorem, that

$$\lim_{n\rightarrow +\infty}{\left(\frac{1}{n} ln\left(\frac{\sum_{i=1}^k{w_ix_i^n}}{x_1^n}\right)\right)}=0$$

 

[hgfalling]

Just think about this simple case.

Suppose you have a function that is M for a little interval of width $\delta$, and zero otherwise. Never mind that it isn't continuous and all that. Now what's the integral there? And what's the limit? That should put you on the right track.