Finally got it, what a lulu.

Use the principle of Archimedes (Ref Rosenlicht or Shilov):

For any real a and integer N, K exists st

K Na< K+1, or

0 a - K/N < 1/N

So we can come to within 1/N of a after K steps.

Mathematically, this also proves that e^iK is dense on the unit circle.