I don't know any simply way, but I would be interested in one, too.
In fact $\sum_{n\geqslant 0}z^{2^n}$ has no radial limit anywhere on the unit circle. This follows from a 1928 Tauberian theorem of Ananda-Rau (see review here). The result is included as Theorem 104 in Hardy: Divergent series (Oxford Clarendon Press, 1948); the proof appears in the notes on Chapter VII.
For part (b) of the problem in Stein-Shakarchi's book, see Theorem 6.4 (on Page 203) in Zygmund: Trigonometric Series I.