- #1
drobadur
- 2
- 0
Hello everyone.
I desperately need clarifications on the least upper bound property (as the title suggests). Here's the main question:
Why doesn't the set of rational numbers ℚ satisfy the least upper bound property?
Every textbook/website answer I have found uses this example:
Let S={x∈ℚ : x≤√2}. This has an upper bound in ℚ, but it has no supremum in ℚ, since √2 is irrational, therefore ℚ does not satisfy the least upper bound property.
Here's my major problem:
A supremum is not necessarily an element of the set it's being a least upper bound of.
By the definition of the least upper bound property:
(i)A real number x is called an upper bound for S if x ≥ s for all s ∈ S.
(ii)A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper
bound of S.
Nowhere does this definition state that the supremum has to belong in that set.
It seems to me that √2 should perfectly qualify as the supremum of S by this very definition. After all x≤√2 for all x∈S.
But why do we dismiss it by saying it is not rational? Why does it HAVE to be? Why does the supremum of a subset of ℚ also HAS to be in ℚ, or else ℚ doesn't satisfy the least upper bound property?
I'm probably missing something and this question looks idiotic to some, but frankly I don't see it. I hope someone can help me out by giving a detailed, comprehensible answer. Thanks in advance.
I desperately need clarifications on the least upper bound property (as the title suggests). Here's the main question:
Why doesn't the set of rational numbers ℚ satisfy the least upper bound property?
Every textbook/website answer I have found uses this example:
Let S={x∈ℚ : x≤√2}. This has an upper bound in ℚ, but it has no supremum in ℚ, since √2 is irrational, therefore ℚ does not satisfy the least upper bound property.
Here's my major problem:
A supremum is not necessarily an element of the set it's being a least upper bound of.
By the definition of the least upper bound property:
(i)A real number x is called an upper bound for S if x ≥ s for all s ∈ S.
(ii)A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper
bound of S.
Nowhere does this definition state that the supremum has to belong in that set.
It seems to me that √2 should perfectly qualify as the supremum of S by this very definition. After all x≤√2 for all x∈S.
But why do we dismiss it by saying it is not rational? Why does it HAVE to be? Why does the supremum of a subset of ℚ also HAS to be in ℚ, or else ℚ doesn't satisfy the least upper bound property?
I'm probably missing something and this question looks idiotic to some, but frankly I don't see it. I hope someone can help me out by giving a detailed, comprehensible answer. Thanks in advance.