I am stuck in the last exercise of my homework, could someone give me a help?
Show that if $f\in PC(2\pi)$and if $x_0,x\in[-\pi,\pi]$, then the Fourier series for $f$ can be integrated term-by-term:
$$\int_{x_0}^xf(t)dt=\frac{1}{2}a_0(x-x_0)+\sum_{n=1}^\infty\int_{x_0}^x(a_n\cos(nt)+b_n\sin(nt))dt$$
In the exercise I am asked to use the following result:
If $k\in PC(2\pi)$ then:
$$\left|\int_{x_0}^xk(t)dt\right|\leq\|k\|_2\sqrt{2\pi}$$
I know that somewhere I must use the Norm Convergence Theorem but I don't know how.