How to prove a number is a supremum of a set

In summary, the conversation discusses proving that if certain conditions are met, then u is equal to the supremum of S. A possible approach to proving this is by contradiction, considering cases where u is greater than and less than the supremum of S and ruling out each possibility.
  • #1
gotmilk04
45
0

Homework Statement


Prove that if
(i) [tex]\forall n[/tex][tex]\in N[/tex], u - (1/n) is not an upper bound of s
(ii) [tex]\forall n[/tex][tex]\in N[/tex], u + (1/n) is an upper bound of S
then, u = supS

Homework Equations





The Attempt at a Solution


It (i) and (ii) are true, then
[tex]\exists s[/tex][tex]\in S[/tex] s.t. u - (1/n) < s
and u+(1/n)>s for all s.
I'm not sure where to go from here.
 
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  • #2


How about a proof by contradiction?

Suppose [itex]u > \sup S[/itex]. Then there exists [itex]n \in \mathbb{N}[/itex] such that [itex]\sup S < u - 1/n < u[/itex]. Does this violate one of (i) or (ii)?

Next suppose [itex]u < \sup S[/itex]. Can you rule this out as well?
 

FAQ: How to prove a number is a supremum of a set

What does it mean for a number to be a supremum of a set?

For a number to be a supremum of a set, it means that the number is the smallest upper bound of the set. In other words, every element in the set is less than or equal to the supremum.

How do I prove that a number is a supremum of a set?

To prove that a number is a supremum of a set, you must show that it is an upper bound of the set and that it is the smallest upper bound. This can be done by comparing the number to every element in the set and showing that it is greater than or equal to each element.

Can there be more than one supremum for a set?

No, there can only be one supremum for a set. This is because the supremum is the smallest upper bound of the set, and if there were more than one, then one of them would not be the smallest.

What if the set has no upper bound?

If the set has no upper bound, then it does not have a supremum. This means that there is no smallest number that is greater than or equal to all the elements in the set.

What if the set is empty?

If the set is empty, then it does not have a supremum. This is because there are no elements in the set to compare to a potential supremum. In this case, the supremum is undefined.

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