# Integral inequality

## Homework Statement

For f nonnegative and continuous on [0,1], prove.
$$\left( \int f \right) ^2 < \int f^2$$
With the limits from 0 to 1.

## The Attempt at a Solution

I was trying to use Upper sums, i.e.
$$\inf \sum \Delta x_i M_i(f^2) = \inf \sum \Delta x_i (M_i(f))^2$$
and then compare this to $$\inf \left[ \sum \Delta x_i M_i(f) \right] ^2$$
Am I in the correct way to prove it?
Why does f is required to be continuous, I didn't use this fact!