Would anyone like to help me complete this proof? I need some help understanding where to go next. The book is giving me hints and I am trying to follow along, but I am getting confused about how to finish.
Let $c$ be irrational with $0
Let $\varepsilon>0$
Ok, so first I prove that $x_n=x_m$ implies $n=m$, which is easy, since $c$ is irrational. So every $x_n$ is unique.
Secondly, I can use the Archimedian property to pick $m$ such that $\frac1m < \varepsilon$ . Then I can divide up the interval $[0,1)$ into $m$ pieces like this: for $1 \leq k \leq m$ I can let $I_k=\left[\frac{k-1}m,\frac km\right)$.
Now I can take $\{{x_j : j=1, N+1, 2N+1,\ldots,mN+1}\}$ , which has $m+1$ distinct values, and thus by the pigeonhole principle, there must be $x_j$ and $x_{j'}$ that are both in the same $I_k$ and hence $|x_j-x_{j'}|<\varepsilon$.
So here I am not sure where to go now. Would anyone care to help me out? I am trying to find the cluster points.