How to Prove an Upper Bound for a Set of Real Numbers?

[matematiker]

Homework Statement

Let A be a set of real numbers. If b is the supremum (least upper bound) of the set A then whenever c<b there exist an a in A such that a>c.

Homework Equations

The Attempt at a Solution

I considered two cases. The first one when the supremum b is attained by the set A. In this case there exists an a belonging to A such that a=b and the statement is proved.

In the second case the supremum is not attained by the set A, so for all a that belong to A, a<b. Here is where I get stucked. I cannot come up with an idea of an a larger than c but smaller than b.

Any hint in the right direction will be very much appreciated. Thank you !

 

[clamtrox]

How about a proof by contradiction? If for c≥a for all points a in A, what does that tell you about the least upper bound?

 

[matematiker]

How about a proof by contradiction? If for c≥a for all points a in A, what does that tell you about the least upper bound?

That would mean b is not the least upper bound since c is smaller than b and greater or equal to any a in A which is a contradiction since by hypothesis b is the supremum of A.

Awesome !
Thank you clamtrox !

 

[matematiker]

That would mean b is not the least upper bound since c is smaller than b and greater or equal to any a in A which is a contradiction since by hypothesis b is the supremum of A.

Awesome !
Thank you clamtrox !

Now I am trying to prove the statement in the other direction:
Let a,b,c be reals and let A be a set of real numbers. If c<b and there is an a in A such that a>c then b is the supremum of A.

From the givens I know that:
c<b
There is an a in A such that a>c.

What I need to prove is that for any a in A, a<=b.
I took a>c from the givens but that is where I get stucked because I do not know how to show that this a is greater or equal to b.

Do you have any hint?
Thank you in advance !

 

[clamtrox]

Let a,b,c be reals and let A be a set of real numbers. If c<b and there is an a in A such that a>c then b is the supremum of A.

There's something wrong here... Let a=1, A={1}, c=0 and b=42. Then c<b and a>c exists, yet b is not the supremum. Maybe it should say "If for all c<b there exists a in A s.t. a>c ... "