I am learning about Fourier Series with Carother's Real Analysis. I'm currently stuck on the bolded statement below made by the author. First I will give a little context.
We have just proved the the partial Fourier sums of $f$ is the closest function to $f$ in the set of trigonometric polynomials of at most degree n. i.e \begin{align} \underset{T \in T_{n}}{inf} ||f - T||_{2} = ||f(x) - S_{n}(f)||_{2} \end{align}
We then used this to prove the following equality: \begin{align} ||f - S_{n}(f)||_{2}^{2} = ||f||_{2}^{2} - ||S_{n}(f)||_{2}^{2} \end{align}
We make the observation $||f - S_{n}(f)||_{2}^{2}\geq 0$ to get Bessel's Inequality: \begin{align} ||S_{n}(f)||_{2}^{2} \leq ||f||_{2}^{2} \end{align}
Because $n$ is arbitrary we can allow $n \rightarrow \infty$. Hence, \begin{align} \frac{a_{0}}{2} + \sum_{k=1}^{\infty}(a_{k}^{2} + b_{k}^{2}) \leq \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)^{2}dx \end{align}
This is the part I get confused at, when the author makes the following statement:
"In particular, the Fourier Coefficients of $f$ must tend to zero: \begin{align} \lim_{n\to\infty}\int_{-\pi}^{\pi}f(x)cos(nx)dx = 0 = \lim_{n\to\infty}\int_{-\pi}^{\pi}f(x)sin(nx)dx \end{align} This fact is know as Riemann's lemma..."
I can't seem to make this jump. I also looked at the following post however, I still couldn't understand it. Thanks for any help, it is much appreciated!
The Fourier coefficients must tend to zero
is a consequence of $$ \frac{a_{0}}{2} + \sum_{k=1}^{\infty}(a_{k}^{2} + b_{k}^{2}) < \infty , $$ which in turn follows from $$ \frac{a_{0}}{2} + \sum_{k=1}^{\infty}(a_{k}^{2} + b_{k}^{2}) \leq \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)^{2}dx , $$ since $f$ is square-integrable. I'm guessing that square-integrablility is assumed somewhere above what you quoted.