1. ## More proofs

1. If E has a supremum, prove that it has only one supremum.

'Was thinking about taking the approach: Suppose E has 2 supremums and prove that it is false. Help on ideas to approach and make this flow will help me lots.

2. Prove that if M is an upperbound of a set E and M is in E, then M is the supremum of E.

Again help on where to go and flow with this one too. I so wish I could wrap my head around these proofs and understand them more. I think having taken all the calc's 15 years ago is part of my problem.

2. Originally Posted by Caity
1. If E has a supremum, prove that it has only one supremum.
Suppose $\displaystyle E$ has more than one supremum and let $\displaystyle s_1$ and $\displaystyle s_2$ be suprema for $\displaystyle E$ with $\displaystyle s_1 \ne s_2$.

Now $\displaystyle s_1$ is a least upper bound for $\displaystyle E$, and as $\displaystyle s_2$ is an upper bound $\displaystyle s_1 \le s_2$.

Now the roles of $\displaystyle s_1$ and $\displaystyle s_2$ in the last paragraph may be reversed and so we conclude that $\displaystyle s_1 \ge s_2$ . Hence $\displaystyle s_1=s_2$ a contradiction, so if $\displaystyle E$ has a supremum it is unique.

CB

3. Originally Posted by Caity
2. Prove that if M is an upperbound of a set E and M is in E, then M is the supremum of E.
Suppose $\displaystyle M$ is not the supremum of $\displaystyle E$, then there is an upperbound $\displaystyle U<M$ for $\displaystyle E$.

But because $\displaystyle U$ is an upperbound for $\displaystyle E$ and $\displaystyle M$ is in $\displaystyle E$; $\displaystyle M \le U$ a contradiction.

CB