However, it is true that any number can be approximated by terms of the sequence with arbitrarily good accuracy: there exists a sequence such that . In other words, the set is dense in .
I think the best way to prove this is to prove a (very slightly) stronger statement about complex numbers: the sequence is dense in the complex unit circle. The result follows by projection (if is close to then is close to , and we take ).
The density of in the unit circle comes from a pigeon-hole principle and from the irrationality of .
The pigeon-hole principle argument is equivalent to the application given in this wikipedia page (paragraph beginning with "A notable problem...") where . I guess you can find other references with the keywords I gave you.
You can find plenty of references about the irrationality of on the web, or just admit it.