The following was extracted from Jeferry's Advanced Engineering Mathematics book.
The Fourier series is given in the following form: $$f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx).$$
I'm curious to why (26) holds. Any help in understanding it is much appreciated.
Here's how: The author says in text that $2a_0 + \sum_{r=1}^{\infty}(a _r^2 + b_r^2)$ is convergent (which is a consequence of Bessel inequality).
Since $a_r ^2 \le a_r ^2 + b_r ^2$ for each $r\in \mathbb N$, we have by Comparison test for series that $\sum_{r=1}^{\infty} a_r ^2$ converges. By the nth term test, we have that $\lim_{r\to \infty} a_r ^2 = 0$. But this implies $\lim_{r\to \infty} a_r = 0$. You can repeat the same argument for $b_r$.
NOTE: I have presumed that $a_r, b_r \in \mathbb R$ for each $n \in \mathbb N$.