Example of a boundary point that is not simple Simple boundary point definition at Planet Math
Rudin gives the following as an example of a boundary point that is not simple:

If $\Omega = U - \{x : 0 < x \le 1\}$ then $\Omega$ is simply-connected. If $0 < \beta \le 1$, $\beta$ is a boundary point that is not simple.

No matter what sequences I try, I can not find a sequence that converges to $\beta$ but a path cannot connect its points. Would you please tell me how this example works?

Try $$x_n=\beta+\frac{(-1)^n}{n} i$$
Is these were connected by a path $\gamma:[0,1)\to\Omega$, then  the real part of $\gamma$ would have to  attain negative values along a sequence converging to $1$.   
