Let $f$ and $g$ be continuous increasing functions defined on $[0,1]$. Prove that $$\left(\int_0^1 f(x)\,dx\right)\left(\int_0^1 g(x)\,dx\right) \le \int_0^1 f(x)g(x)\,dx$$
This looks like Cauchy-Schwarz, using $L^1$(?) but that makes little sense to me. Perhaps there is a more elementary way of proving this inequality?
Note: This question is from a undergraduate single variable calculus course, but feel free to use any suitable theorems from analysis.
That is just the continuous version of Chebyshev's sum inequality, which on its turn is a straightforward consequence of the rearrangement inequality.
The continuous version follows from the discrete version by considering Riemann sums (if $\int$ stands for a Riemann integral) or approximations through simple functions (if $\int$ stands for a Lebesgue integral).