How to prove $(\space|\sin n|\space)$ does not converge? How do we prove that $(\space|\sin n|\space)$ is not convergent ? There is a beautiful proof of the non-convergence  of $(\sin n)$ by considering the identities $\sin (n+1)=\cos1\sin n+ \sin 1 \cos n $ , 
$\cos (n+1)=\cos1\cos n- \sin 1 \sin n $ and $ \sin^2 n+\cos^2n=1$ , is there a similar proof for $(\space|\sin n|\space)$ ? (though at least any kind of proof will be helpful) 

Hint: We know $\sin(x)$ and $|x|$ are continuous functions, and so $|\sin(x)|$ is a continuous function. Can you find this limit $\lim_{x\rightarrow \infty}|\sin(x)|$? It does not exists. Do you want to see the proof? Try to apply Cauchy's condition of infinite limite.
Apply the sequential criterion of continuity and see $|\sin(n)|$ does not exists. 
