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Find a sequence $\{a_n\}$ in $B(0,1)$ such that $\sum(1-|a_n|)<\infty$ and every number $e^{i\theta}$ is a limit point of $\{a_n\}$.

This is an exercise from Conway that I am stuck at. What is an example of such a sequence? I thought of some spiral shaped sequence, but cannot give a rigorous construction of such a sequence. Could anyone please help me?

Robert Z
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Keith
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1 Answers1

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Hint. Take a look at Density of $e^{in \alpha}$ and try with the spiral shaped sequence $$a_n:=\left(1-\frac{1}{2^n}\right)e^{in}\quad \text{for $n\geq 0$}.$$

Robert Z
  • 147,345