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?
Best 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 $-\infty$.