Sequence Limit: $\sin(n^3)$ How can I prove that this sequence does not converge, using  the definition?

$$W_n = \sin(n^3)$$

For $n \in \mathbb{N}$. I tried to do a proof by reduction to the absurd but without result.

Reference: https://terrytao.wordpress.com/2010/03/28/254b-notes-1-equidistribution-of-polynomial-sequences-in-torii/
As a special case of Corollary 6, we have

Let $P(n)=a_s n^s + \cdots + a_0$ be a polynomial with real coefficients. If $a_s$ is irrational, then $P(n)$ mod $1$ is equidistributed. 

With $P(n)=\frac1{2\pi} n^3$, we have $P(n)$ mod $1$ is equidistributed. Thus, $n^3$ mod $2\pi$ is equidistributed. Then it follows that $\sin n^3$ is dense in $[-1,1]$. 
