1. ## Supremum Bounds

How can there be more than 1 upper bound? Can't you use the same argument for showing that there is only 1 greatest lower bound?

That is, inf(A) is unique. Suppose s_1 and s_2 are both inf(A). Since s_1 is a greatest lower bound, s_2 <= s_1.

But, since s_2 is a greatest lower bound, s_1 <= s_2

This implies that s_1 = s_2.

Can't this argument be used for upper bounds...

Is it because of the notion that if you pick some "upper bound" you can always find something that's higher?

2. Originally Posted by seerTneerGevoLI
How can there be more than 1 upper bound? Can't you use the same argument for showing that there is only 1 greatest lower bound?

That is, inf(A) is unique. Suppose s_1 and s_2 are both inf(A). Since s_1 is a greatest lower bound, s_2 <= s_1.

But, since s_2 is a greatest lower bound, s_1 <= s_2
Let $\displaystyle S$ be a non-empty bounded set of real numbers. By the completness property we have that there exists $\displaystyle s_1$ which is a least upper bound. Say there is another least upper bound $\displaystyle s_2$. Now $\displaystyle s_2$ is an upper bound. Thus, $\displaystyle s_1\leq s_2$ because $\displaystyle s_1$ is least. Similarly, $\displaystyle s_2\leq s_1$. Thus $\displaystyle s_1=s_2$. Q.E.D.