Fourier Coefficients of a Sequence of Functions

Question

Let $f_k$ be a sequence of Riemann integrable functions over $[0,2\pi]$ such that $$\lim_{k\rightarrow\infty}\int_0^{2\pi}|f_k-f|=0$$ for some function $f$. Let $\hat{g}(n)$ denote the $n$th Fourier coefficient of $g$. Prove that for all $n$ $$\lim_{k\rightarrow\infty}\hat{f_k}(n)=\hat{f}(n)$$

Answer 1

Start with $$ \hat f(n) - \hat {f_k}(n) = \frac1{2\pi} \int_0^{2\pi} \big(f(x) - f_k(x)\big)e^{-inx} \, dx$$ and apply the integral version of the triangle inequality.