Proving convergence by proving a sequence does not diverge Is it possible to prove convergence by proving that a sequence does not diverge? 

Especially I don't know how to deal with periodic sequences such as $\lim \limits_{n \to \infty } \sin(n)$.
By definition any sequence that does not converge, diverges. Thus, by contraposition, if a sequence does not diverge, it must converge. So technically, yes, you could do that. I am not sure how useful this would actually be.

In this particular case your limit diverges, so it does not really apply here.
your limit diverges? i think it must be the given sequence, the limit doesn't exist