Problem: $\underset{N\to\infty}{\lim}\int_{-\pi}^\pi [g(t)\cos(\frac{t}{2})]\sin(Nt)dt=0$, where $g(t)=\frac{f(x-t)-f(x)}{\sin(t/2)}$.
The Problem arises from Walter Rudin's PMA:
For the texts highlighted in blue: the boundedness of $g(t)\sin(t/2)$ is trivial; and to show the boundedness of $g(t)\cos(t/2)$, I suppose we show that $\underset{t\to 0}{\lim}\frac{t}{cos(t/2)}=2$, then combine it with (79) and (81). But the texts highlighted in yellow really trouble me. (74) states that "$\underset{n\to\infty}{\lim}c_n=0$", where $c_n$ is the n^th Fourier coefficient of $f$ relative to an orthonormal series ${\phi_n(x)}$. Here I believe $c_n=\int_{-\pi}^\pi f(t)\overline{\phi_n(t)}dt$ by (66). But then how does that lead to "$\int_{-\pi}^\pi [g(t)\cos\frac{t}{2}]\sin Nt dt$ and $\int_{-\pi}^\pi [g(t)\sin\frac{t}{2}]\cos Nt dt$ tend to $0$ as $N\to\infty$"?
Any hint would be greatly appreciated.