I'm asked to show that there cannot be $\alpha_1,\alpha_2,...\in\mathbb{C}$ s.t.
$$\lim_{N\to\infty}\int_{-\pi}^{\pi}|e^{it}-\sum_{k=1}^{N}a_k\sin(kt)|^2dt=0$$
Here is my attempt:
Assume there are such $\alpha_1,\alpha_2,...\in\mathbb{C}$. The limit being $0$ would imply the convergence of $s_j=\sum_{k=1}^{j}\alpha_k\sin(kt)$ to $e^{it}$ in $\mathbb{L_2}(-\pi,\pi)$. If we now write $\beta_j=\frac{\alpha_j}{\sqrt\pi}$ we have $s_j=\sum_{k=1}^{j}\beta_k\frac{\sin(kt)}{\sqrt\pi}$. And since $\{\frac{\sin(kt)}{\sqrt\pi}\}_{k\in \mathbb{N}}$ is an orthonormal sequence in $\mathbb{L}_2(-\pi,\pi)$ we know that we must have $\beta_k=<e^{it},\frac{\sin(kt)}{\sqrt\pi}>$. And after some work one find that $\beta_1=i\sqrt\pi$ and all other $\beta_i=0$. So that $s_j=\sum_{k=1}^{j}\beta_k\frac{\sin(kt)}{\sqrt\pi}=i\sin(t)\neq e^{it}$ which establishes the contradiction.
I think this is correct but I would like to be sure. Thanks