I had encountered proof of above problem here.Dense set in the unit circle- reference needed
Here I do not understand following arguments.
1) Even if $S^1$ has finite length for any $\epsilon >0$ how to find $n_1,n_2$ such that $|e^{in_2x}-e^{in_1x}|<\epsilon$ ?
2) How for any $\psi \in \mathbb R/\mathbb Q$ that sequnce obtained converges to to$\psi$.
Any Help will be appreciated
1) is a pigeon hole argument: If you devide the circle for given $\epsilon$ into disjoint arcs of length $<\epsilon/\pi$ one of them contains two of the (infinitely many) points $e^{inx}$, say for $n_1<n_2$. Then note that the eucledian distance between two points on the circle is less or equal than $\pi/2$ times the arclength between the points.
2) This does not directly give a convergent sequence but the following statement: For each $\epsilon>0$ and $\zeta\in S^1$ there is $n_\epsilon$ with $|e^{in_\epsilon x}-\zeta|<\epsilon$. To get a convergent sequence apply this for each $\epsilon=1/k$.
