Problem: Let $\Omega=(0,1),\,\mathcal F=$ Borel sets, $P=$ Lebesgue measure. Define the sequence of random variables $X_n(\omega)=\sin(2\pi n\omega),\,n=1,2,\dots$. Show that the random variables are uncorrelated but not independent.
Attempt: Let $n,m\in\mathbb N$ with $n\ne m$.
\begin{align}
E[X_nX_m]&=\int_0^1\sin(2\pi n\omega)\sin(2\pi m\omega)\,dP\\
&=\frac{1}{2}\int_0^1\cos[2\pi \omega(n-m)]-\cos[2\pi\omega(n+m)]\,dP\\
&=\frac{1}{2}\left[\frac{\sin[2\pi\omega(n-m)]}{2\pi(n-m)}-\frac{\sin[2\pi\omega(n+m)]}{2\pi(n+m)}\right]\Bigg\vert_0^1\\
&=0.
\end{align}
Next, we have
$$E[X_n]=\int_0^1\sin(2\pi n\omega)\,dP=-\frac{\cos(2\pi n\omega)}{2\pi n}\Bigg\vert_0^1=0.$$
It follows that
$$E[X_nX_m]-E[X_n]E[X_m]=0.$$
Therefore, the random variables are uncorrelated.
Now we prove that they are not independent. Let $n,m\in\mathbb N$ with $n\ne m$. Then $X_n(\omega)=0$ for $\omega=k/2n$ where $k\in\mathbb N$ with $0\leq k\leq 2n$. On this set, $X_m$ takes on
the values $\{y_0,y_1,\dots,y_{2n}\}$. Now define $V_m=\bigcup_{j=0}^{2n}(y_i,y_i+\varepsilon)$, where $\varepsilon>0$ is sufficiently small. Now let $a<b$ with $[a,b]\subset[0,1]\setminus V_m$.
The continuity of the sine function implies that
$$P(X_n\in[0,\varepsilon],X_m\in[a,b])=0<P(X_n\in[0,\varepsilon])\cdot P(X_m\in[a,b]).$$
It follows that the random variables are not independent.
I am concerned about my work for the proof that the random variables are not independent. I graphed some examples and it seems to work, same as for the actual proof, but I have a hunch that it is not airtight.
Could anyone please give me some feedback on the proof above and whether it is airtight or not? Any comments are most welcomed and appreciated.
Thank you very much for your help.