ok so i need to prove this without using least upper bounds
Prove an increasing sequence, bounded above, is convergent.
I'm not sure how to start thanks!
Why do you need to prove it without using least upper bounds?
I suppose you could do it by showing that it is a Cauchy sequence and using the fact that Cauchy sequences converge.
Other than that, you could try looking at all bounded sequences of rational numbers that have an upper bound. If you say that two such sequences, $\displaystyle {a_n}$ and $\displaystyle {b_n}$ are "equivalent" if and only if the sequence $\displaystyle {a_n- b_n}$ converges to 0, then you can DEFINE the real numbers to be the equivalence classes. That way, the theorem follows immediately.
One difficulty with not showing any work at all is that we cannot know what methods you are familiar with or what definitions you are using. There are many different ways to prove any specific theorem!
I was thinking about using Cauchy but I was a little confused. For instance, if the sequence is a_n, we know there exists a convergent subsequence a_m_n. so a_m_n is cauchy.
say a_m_n converges to L.
Then
|a_n - a_m_n+a_m_n - L | <= |a_n - a_m_n| + |a_m_n - L |
I need to make the first part |a_n - a_m_n| small. But a_n might not be a part of the subsequence a_m_n and so I don't know how to proceed.
Thanks for the posting tips.
Our professor specified not to use least upper bound. I think she's trying to help us be more creative with our proofs.