First one should specify what "reach" means: there are (many) real number

such that

for all

(just because

is uncountable).

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.