Prove that all real sequences, $(x_n)$, which are increasing have a limit. Prove that all real sequences, $(x_n)$, which are increasing have a limit
I figured a decent way to ready for my Real Analysis class next semester would be to step through a book and try to prove the theorems for myself.  I didn't realize how difficult some of them would be.  :(
I believe this one should be broken up into two cases.
Case 1:  The sequence is bounded above.  I believe there is a theorem I already proved that covers this case.
Case 2:  The sequence is not bounded above.  I could prove this through contradiction (much like the natural numbers are not bounded above).  In this case the limit diverges to infinity.  Or, in the extended reals (I believe) the limit would be infinity.
Thoughts on this?  Thank you for any help.
p.s.  If there is a question that already has this answer and you post it, please tell me how you found it.  I'm looking through "Questions that may already have your answer" and didn't see any that were similar.
 A: For this to be true you would need to interpret "having a limit" to allow this limit to be infinity - because that's what is going to happen if the sequence is unbounded. I don't really see how you'd prove this by contradiction as the general idea in both cases would be the same.
For the bounded case we would be using the axiom of least upper bound. This will imply there's a least upper bound $M$ for the sequence. As beeing the least then for every $\delta>0$ we have some $\omega$ such that $x_\omega > M-\delta$. Since the sequence is increasing we have for every $j>\omega$ that $M\ge x_j>M-\delta$. This means by definition that $x_j\to M$ as $j\to\infty$.
For the unbounded case the idea is the same. Since it's unbounded then for every $L$ there's a $\omega$ such that $x_\omega>L$ and since the sequence is increasing we have for every $j>\omega$ that $x_j>L$. This means by definition that $x_j\to\infty$ as $j\to\infty$
A: Yes, this is true. If you want to prove Case 2, you really should prove it properly. Proving the contrapositive (or by contradiction) is a good way to go.
