I'm having a trouble with the exercise 16 in chapter 4 of Stein's "Real Analysis, Measure Theory, Integration, Hilbert spaces".
Let $F_0(z) = 1/(1-z)^i.$
(a) Verify that $|F_0(z)|\le e^{\pi/2}$ in the unit disc, but that $\displaystyle\lim_{r\to1}F_0(r)$ does notexist.
[Hint: Note that $|F_0(r)| = 1$ and $F_0(r)$ oscillates between $\pm1$ infinitely oftenas $r\rightarrow1$.]
(b) Let $\{a_n\}_{n=1}^\infty$ be an enumeration of the rationals, and let$$F(z) = \sum_{j=1}^\infty\delta^jF_0(ze^{-i\alpha_j})$$where $\delta$ is sufficiently small. Show that $\displaystyle\lim_{r\to1} F(re^{i\theta})$ fails to exist whenever $\theta = \alpha_j$, and hence F fails to have a radial limit for a dense set of points on the unit circle.
(a) is OK, but I'm having trouble with (b). It looks to me that $F(z)$ does NOT fail to exist. Here's what I attempted.
When $\theta = \alpha_k$ for some $k \in \Bbb N$,$$F(z) = \delta F_0(re^{i(\alpha_k-\alpha_1)}) + \delta^2 F_0(re^{i(\alpha_k-\alpha_2)}) +... + \delta^k F_0(r) + \delta^{k+1}F_0(re^{i(\alpha_{k+1}-\alpha_1)})+...$$where all terms including $k^{th}$ term will vanish as $r\to 1$ because by (a) all $F_0$ is bounded and $\delta^j$ is sufficiently small regardless of the oscillation in $k^{th}$ term.
Could someone help me? Thank you in advance!