Comparing two different weighted averages of a function.

Question

I need to prove the following:

Assumptions:

1. $f(\cdot)$ and $g(\cdot)$ are continuous and weakly increasing on $[0,1]$.
2. $f(x)\geq 0$ and $g(x)\geq 0$ for $\forall x \in [0,1]$.
3. $\int_0^1g(x)dx = 1$.

Statement: $\int_0^1 f(x)g(x) dx \geq \int_0^1 f(x) dx.$

I think the statement is true because the left hand side of inequality is a weighted average value of $f$ with more weight put on the 'right' side of $f$ and $f$ is increasing, but I have a hard time writing a proof.