1. ## sup/inf

Hi,
I understand the basic definition of sup and inf, but when it comes to proving the sup or inf of an infinite set where infS/supS is not an element of S, I am having trouble.

For example, given a problem of proving that

supS=0 where S={x|x<0},

I am able to prove the first part of the definition of supS (proving that 0 is an upper bound of S), but I always get stuck at the 2nd part of the definition

(proving that for any b, where b is an upper bound of S, 0 is less than or equal to b).

Any help would be appreciated

2. Originally Posted by lexietx
Hi,
I understand the basic definition of sup and inf, but when it comes to proving the sup or inf of an infinite set where infS/supS is not an element of S, I am having trouble.

For example, given a problem of proving that

supS=0 where S={x|x<0},

I am able to prove the first part of the definition of supS (proving that 0 is an upper bound of S), but I always get stuck at the 2nd part of the definition

(proving that for any b, where b is an upper bound of S, 0 is less than or equal to b).

Any help would be appreciated
Suppose that $\alpha$ is a lower bound for $S$ then $\forall x>0,\alpha ?

3. Assume to the contrary that for there exists some bound b, 0 > b. Well, if b is a bound, then we know for b>= all the elements of the set. But then 0 is a greater bound than the bound b, so 0 cannot be the least upper bound, so Sup =/= 0, which is a contraction.