Calculus of a Single Variable, Chapter 9, 9.1, Section 9.1, Problem 43

Theorem 1
Let a in RR then lim_(n to infty)a/n=0.
We will also use the fact that the codomain of sine function is [-1,1].
lim_(n to infty)a_n=lim_(n to infty)sin n/n=
We cannot calculate sin infty, but we know that its value must be between -1 and 1. Therefore, using theorem 1 we get
lim_(n to infty)sin n/n=0
As we can see the sequence is convergent and its limit is equal to 0.
The image below shows first 100 terms of the sequence. We can see that the sequence is alternating, but the terms are slowly approaching x-axis i.e. the sequence converges to zero.