What is $\sup(\sin(n))$? My classmate asked a question during lecture about our discussion of bounded sequences, particularly the sequence $\sin(n)$. His question was, What is $\sup(\sin(n))$?

The set of points of the form $\sin(n)$, where $n$ ranges over the integers, is bounded above by $1$. It turns out that $1$ is the supremum of the set.
One can show that the numbers of the form $\sin(n)$ are in fact dense in the interval $[-1,1]$. So for any number $x$ in that interval, there are integers $n$ such that $\sin(n)$ is arbitrarily close to $x$.
There is no integer $n$ such that $\sin(n)=1$. However, there is an integer $n$ such that $\sin(n)\gt 0.99$. There is also an integer $n$ such that $\sin(n)\gt 0.9999$. And so on.   
