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 $E$ has more than one supremum and let $s_1$ and $s_2$ be suprema for $E$ with $s_1 \ne s_2$.

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

Now the roles of $s_1$ and $s_2$ in the last paragraph may be reversed and so we conclude that $s_1 \ge s_2$ . Hence $s_1=s_2$ a contradiction, so if $E$ 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 $M$ is not the supremum of $E$, then there is an upperbound $U for $E$.

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

CB