Proving the Least Upper Bound Property: A Mathematical Inquiry

[ssayan3]

Least Upper Bound proof...

Homework Statement

Suppose A is a nonempty set that has x as an upper bound. Prove that x is the least upper bound of the set A iff for any E>0 there exists a y in A such that y>x-E

Homework Equations

None

The Attempt at a Solution

The forward where you assume that x is the least upper bound is very easy, but I'm having some trouble proving the reverse...

This is what I have so far...

Let x be an upper bound of A, and choose a point z in A.
If x is an upper bound of A, then x+z is also an upper bound.

 

[LCKurtz]

To prove the reverse, you are given that x is an upper bound for A having the property:

If \epsilon &gt; 0 there is a y in A satisfying x - \epsilon < y

You have to show that no number z < y is an upper bound for A. What problem would arise if there was such a number z?

[Edit] Sorry, there is a typo. The last paragraph should have read:

You have to show that no number z < x is an upper bound for A. What problem would arise if there was such a number z?

 

[ssayan3]

Hmm... if there were such a number z, then y could not be the least upper bound...

Could the proof go something like this?:

Choose arbitrary E>0, and let y be an upper bound of A

Suppose z is an upper bound of A, and y>z>y-E.

y is not the lub

does this finish the proof?

 

[LCKurtz]

Hmm... if there were such a number z, then y could not be the least upper bound...

Could the proof go something like this?:

Choose arbitrary E>0, and let y be an upper bound of A

Suppose z is an upper bound of A, and y>z>y-E.

y is not the lub

does this finish the proof?

No. Sorry, but I had a typo which I have corrected. Read my reply and try again.