Infinite sum of square root I tried demonstrate that $\displaystyle \sum_{k=0}^\infty\sqrt{a_n^2+b_n^2}$ is convergent with $a_n$ and $b_n$ are the fourier coefficients of continuos $f(x)$ with $f(x)\in L_2(-\pi, \pi)$ and $f(\pi)=f(-\pi)$. My attempt to show it is to proof that the sum $$\sum_{k=0}^\infty a_n^2+b_n^2$$ is convergent (using Bessel inequation). But I'm not sure if $$\sum_{k=0}^\infty\sqrt{a_n^2+b_n^2}\leq \sum_{k=0}^\infty a_n^2+b_n^2$$
or is it not necessary?

The inequality holds in the opposite direction with minor changes. Since $\lim_{n\to \infty}a_n=\lim_{n\to \infty}b_n=0$ for $f(x)\in L_2(-\pi,\pi)$, we have some $M$ such that $$n\ge M\implies \sqrt{a_n^2+b_n^2}<1$$hence
$$
\sum_{n=M}^\infty {a_n^2+b_n^2}<\sum_{n=M}^\infty \sqrt{a_n^2+b_n^2}
$$
