TITLE: Sequences unbounded above diverges?
QUESTION [0 upvotes]: Prove or disprove (with counterexample) the following statement:
If $a_n > 0$ for all $n \in \mathbb{N}$, and $(a_n)_{n \in \mathbb{N}}$ is not bounded from above, then $a_n \to \infty$.

I thought the answer was true, since if the sequence is unbounded from above, and $a_n$ is greater than $0$, then it'd constantly increast to infinity. But it looks like the answer is false, and I don't really see why. What's a counterexample that refutes this statement? Thank you.

REPLY [0 votes]: Consider the sequence $a_n=n(1+(-1)^n)$ which oscillates unboundedly,but $a_n$ does not tend to $\infty$.
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