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My question is: Does every strictly increasing sequence converge or tend to infinity? I can't seem to find a counterexample, however, I can't seem to be able to prove this either.

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Yes, if it's increasing then the limsup = the limit; the limsup always exists and is either finite or $\infty$. –  mjqxxxx Sep 30 '12 at 15:09
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up vote 3 down vote accepted

Yes.

Suppose $a_n$ increases ($a_{n+1}\ge a_n$ is enough). Then either it is bounded, or not. If yes, then it converges to the $\sup a_n$, else it goes to $+\infty$.

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