# If a monotone increasing sequence is not convergent, then is the sequence not bounded?

If a monotone increasing sequence is not convergent, then is the sequence unbounded?

I know that a monotone increasing sequence that is convergent must be bounded.

But $P \to Q$ being true doesn't necessarily mean $-P \to -Q$ must be true.

• This is the "completeness axiom" for the real numbers Feb 19, 2017 at 22:37
• A monotone increasing sequence is convergent if and only if it is bounded above
– joeb
Feb 19, 2017 at 22:40
• @joeb And below too $\ddot \smile$ Feb 19, 2017 at 22:40
• @user3000482 What is $P$ and what is $Q$ to you? Feb 19, 2017 at 22:41
• Consider rational numbers $1, \frac75, \frac{41}{29}, \ldots$ where if a term is $q$ the next is $\frac{4+3q}{3+2q}$. This is monotone increasing and is clearly bounded above by $2$, but does not converge to a rational number Feb 19, 2017 at 22:52

Let $(a_n)$ an increasing and divergent sequence of real numbers.
The monotone convergence theorem states that if $$(a_n)$$ is monotone then it converges if and only if it is bounded.