Sine function dense in $[-1,1]$

The hard part is to show that for any $x$ such that $0 \le x \le 2\pi$, and any $\epsilon>0$ there exists a real number $y$ and two integers $m$ and $n$ such that $|y-x|<\epsilon$ and $n=2\pi m+y$. Hint: break up $[0,2\pi]$ into small subintervals, remember that $\pi$ is irrational and apply the pigeonhole principle.

Jmoy's is a good hint, but I think an even nicer approach is to prove that the points $(\cos n,\sin n)$ are dense on the unit circle, and then deduce the results for the individual terms.

Try proving that for any irrational number $\alpha$, the set $A=\left \{ a+b\alpha \mid a\in \mathbb{N},b\in \mathbb{Z} \right \}$ is dense in $\mathbb{R}$. Let $\alpha =\pi $. Since the set $A$ is dense in $\mathbb{R}$, $\forall \; x\in \mathbb{R}$, there exists a sequence of terms $(z_{n})$ such that $\lim_{n\rightarrow \infty }\left ( z_{n} \right )=x$.

$(z_{n})$ can be written as the sum of two sequences of integers.

$z_{n} = x_{n} + \pi y_{n}$.

Use the continuity of $\sin(x)$, to show that a sequence from the set $\left \{\sin (n)\mid n\in \mathbb{N} \right \}$ converges to every element of $[-1,1]$.