subsets of euclidean space
- Thread starter sayy
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pka
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“show that every bounded set in R has a least upper bound.”
Of course the empty set is bounded and does not have a least upper bound.
The completeness axiom states that: every non-empty bounded set in R has a least upper bound.
One does not show or prove an axiom.
What may happen is that there is a different axiom and one shows they are equivalent. What axioms are you given?
Of course the empty set is bounded and does not have a least upper bound.
The completeness axiom states that: every non-empty bounded set in R has a least upper bound.
One does not show or prove an axiom.
What may happen is that there is a different axiom and one shows they are equivalent. What axioms are you given?
It is not necessary to have an axiom for the least upper bound property of the real numbers.pka said:
In Principle of Mathematical Analysis, Rudin constructs the real numbers from the rationals and proves that the real numbers constitute a field with the least upper bound property. All of this is done only with set theory and the field properties of Q.
In any case (to repeat pka's question), what theorems or axioms about the real numbers do you know already?