Show that for any $f,\,g: \mathbb{R}\rightarrow\mathbb{R}$ where $f^2$ and $g^2$ are integrable on some interval $I\subset \mathbb{R}$,
$$\left(\int_{I} f(x)g(x)\, dx\right)^2 \leq \int_{I} f^2(x)\, dx \int_{I} g^2(x)\, dx$$
I know that for any $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$, $(\mathbf{x}^T\mathbf{y})^2 \leq (\mathbf{x}^T\mathbf{x})(\mathbf{y}^T\mathbf{y})$, the Cauchy-Schwarz inequality in the L2 norm, and I have figured out its proof. But I can't figure out how to prove the case where we are integrating over $x$.
Rearrange $\int_I(f-\lambda g)^2dx\ge0$ for $\lambda=\frac{\int_Ifgdx}{\int_Ig^2dx}$.