Everywhere divergent trigonometric series (Q1100676)

The author proves the following theorem: Let \((a_ n)\) be a decreasing sequence converging to 0, for which \(\sum^{\infty}_{n=0}a_ n=\infty\). Then there exists a sequence \((e_ n)\) with \(e_ n=0\) or 1 such that the series \(\hat{\;}\sum^{\infty}_{n=0}a_ ne_ n\cos nt\) is divergent everywhere in R.