Past exam of the real and harmonic analysis course of the University of Oxford / 2007 / 3 / c / Solution

Write $A_N(f)=(N+1{)}^{-1}{\sum }_{n=0}^{N}f(e^{in\xi })$. For a monomial $z^k$, $k\in \mathbb{Z}$, the geometric series givesThus every such average converges, and by linearity so does the average of every trigonometric polynomial. Also $|A_N(f)|\le \Vert f{\Vert }_{\infty }$ uniformly in $N$.

Given $\epsilon >0$, choose a trigonometric polynomial $P$ with $\Vert f-P{\Vert }_{\infty }<\epsilon$. ThenThe last term tends to zero as $N,M\to \infty$. The scalar sequence is Cauchy and therefore the limit exists for every continuous $f$ and every real $\xi$.