Let $\mathbf S = \sin(n)$, where $n \, \epsilon \, \Bbb N$. Prove $ \sup \mathbf S = 1$ and $ \inf \mathbf S = -1$.
I understand that the $\sin$ function for real entries is normally bounded by $1$ and $-1$, but I am at a loss for how to translate that to real analysis to prove this.
An idea-- show that the natural numbers contain elements that are arbitrarily close to multiples of pi/2 by odd numbers. Find an element close to a multiple by an odd number with remainder 1 mod 4 for supremum and and by an odd number with remainder 3 mod 4 for infimum. Haven't taken real analysis yet so not sure about this, btw. Similar ideas might show that any periodic function has the same supremum and infimum on the natural numbers as on the reals.