Show that $\{e^{in}: n\in\Bbb N\}$ is Dense in the Unit Circle

This problem gave me fits when I was in grad school. Looking back at it now, it still escapes me. The problem is from Conway's Functions of One Complex Variable. I'm looking for a proof from basic principles, so no big theorems please.

I remember that the Pigeonhole principle was involved, but I don't seem to be able to formulate it in the proper way.

Let $S$ be the unit cirlce, and $T=\{e^{in}:n\in\Bbb N\}$. For a fixed $k$, $S$ can be paritioned into $$A_{m,k}=\left\{e^{i\theta}:\frac{2\pi i(m-1)}{k}\le\theta\le\frac{2\pi im}{k},m=0,\dots k-1\right\}$$. Now infinitely many members of $T$ must belong to $A_{m,k}$. Then $A_{m,k}$ can be partitioned in a similar manner, and in this way we can create a sequence of sets $S_n$ such that $S_{n+1}\subset S_n$ each of which contains infinitely many points of $T$.

So, $\cap_{n=1}^{\infty}S_n$ is not empty, and since $\rm diam\, S_n\to 0$, then intersection is a singleton, which is a limit point of $T$. But this doesn't help, we already know $T$ has limit points in $S$ because it is infinite and $S$ is compact.

Solution 1:

This is identical to showing that the fractional parts of $\dfrac{n}{2\pi}$ are dense in $[0,1)$, which is true since $\dfrac{1}{2\pi}$ is irrational (see e.g. this question).