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.