## 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.

## A cautionary Mathematica tale

Because a picture’s worth a thousand words: according to Mathematica, the rank of a matrix and that of a basis for its range can differ

The matrix involved is a projection, not some numerically unstable thing like the Hilbert matrix. You suck at linear algebra, Mathematica.