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I recently learned the following theorem about bounded sequences:

If a sequence is eventually increasing and not bounded above, then it is divergent to positive infinity.

If a sequence is eventually decreasing and not bounded below, then it is divergent to negative infinity.

Do I have to say "eventually increasing" / "eventually decreasing" when stating this theorem? What if I just say:

If a sequence is not bounded above, then it is divergent to positive infinity.

If a sequence is not bounded below, then it is divergent to negative infinity.

Am I correct in assuming that the sequence being eventually increasing/decreasing is implied by the sequence not being bounded? Are there any drawbacks to the more concise statement that I may not be aware of?

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    $\begingroup$ You can prove something weaker, you can show that if a sequence is not bounded above then a subsequence is divergent to positive infinity. $\endgroup$
    – Yanko
    Aug 8, 2019 at 13:51
  • $\begingroup$ A sequence like $a_n = n + (-1)^n(n/2)$ diverges to $+\infty$ but is not "eventually increasing". $\endgroup$
    – alephzero
    Aug 8, 2019 at 22:43

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You have to say "eventually increasing" or "eventually decreasing".

Consider the sequence $a_n = (-1)^n n$

It is definitely not bounded (above or below) but it doesn't diverge to $\infty$ nor does it diverge to $-\infty$.

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    $\begingroup$ My other favorite example is $n\sin(n)$. $\endgroup$
    – Randall
    Aug 8, 2019 at 13:48

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