Let $$ T:\mathbb{S}^1\to\mathbb{S}^1\\ x\mapsto 2x $$ be the angle-doubling map on the circle. We know that this transformation is ergodic. We want to prove that is mixing.
I have to show that $$ \lim_{n\to\infty}\int_X f\circ T^n g\mathrm{d}m = \int_X f\mathrm{d}m\int_Xg\mathrm{d}m $$ for every $f,g\in L^2$.
The idea is to consider $e^{imx}$ which spans $L^2$. Then I can't go on.
Two standard arguments:
(1) In order to show this equation holds for all $f,g \in L^2$, it suffices to prove it for all $f,g$ in some dense subset $V \subseteq L^2$. This is just a continuity argument (one should be a bit suspicious of the limit on the left, but it's not hard to deal with since $T$ is unitary).
(2) Since both sides are bilinear in $(f,g)$, if we can show this equation holds for all pairs $f, g$ in some collection $\lbrace v_1, v_2, \dots \rbrace$, then it also holds for all pairs in the span $V$ of the $v_i$.
As a consequence of (1) and (2), and the fact that the $e^{imx}$ span (a dense subspace of) $L^2$, it suffices to show that your equation holds for $f = e^{ikx}, g = e^{imx}$, where $k, m \in \mathbb{Z}$.
Now it's just a matter of plugging in and checking. (Deal separately with the cases $k = 0$ and $k \neq 0$, and remember that $\int_X e^{ilx}$ is $1$ if $l = 0$, and zero otherwise.)