A sequence whose closure is the unit circle

Looks like I’ll be visiting U. Michigan for a week or so at the end of October. What does one do on such a visit? I’ve been invited to give a talk, but that doesn’t occupy more than an afternoon…

I spent this morning reading through some of the literature on subspace tracking, then wound up visiting Nocedal and Wright for a refresher on the augmented Lagrangian method (Chap 17). Flipping through the book, I came across the following question:

Show that every point on the unit circle is a limit point of the sequence
\vec{x}_k = \left(1 + \frac{1}{2^k}\right) \begin{pmatrix} \cos k \\ \sin k \end{pmatrix}.

Not challenging (esp. if you don’t go into the nasty \(\varepsilon-\delta\) details ), but it’s a cute problem.