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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 $\lim \limits_{n \to \infty } \sin(n)$.

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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.

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your limit diverges? i think it must be the given sequence, the limit doesn't exist

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