# Can I say "if a sequence is not bounded above, then it is divergent to positive infinity" without explicitly saying it's eventually increasing?

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?

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

## 1 Answer

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 $$-\infty$$.

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