Least Upper Bound proof

  • Thread starter ssayan3
  • Start date
  • #1
15
0
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.
 

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,559
770


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

If [itex]\epsilon > 0[/itex] there is a y in A satisfying x - [itex]\epsilon[/itex] < 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?
 
Last edited:
  • #3
15
0


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?
 
  • #4
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,559
770


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.
 

Related Threads on Least Upper Bound proof

  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
4
Views
5K
  • Last Post
Replies
4
Views
1K
Replies
4
Views
6K
  • Last Post
Replies
11
Views
5K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
11
Views
1K
Top