Real analysis – Basics III
— 3. Completeness Axiom —
As was said in the previous post the description of the set of real numbers made so far allows one to do almost everything but to formally differentiate the set and the set . To do that we need one more axiom; but first let us introduce some auxiliary notions.
For one usually writes , , and for the real numbers for which the following conditions are verified:
is called a closed interval and is called an open interval.
We can also have another kind of intervals. These intervals are said to be infinite and have the form , , , and . They represent respectively:
Definition 6 Let . If , then is the maximal element of and we denote it by
If , is said to be the the minimal element of and is denoted by . |
Definition 7 Let and the set of all of its upper bounds ( iff isn’t bounded from above). The minimal element of , , is said to be the supremum of . Formally: |
The notion of the infimum of a set can be introduced in an analogous way.
Definition 8 Let and the set of lower bounds of . The infimum of is the maximal element of . If such a a maximal element exists the infimum of is denoted by and is the real number such as: |
Let us see some examples of all of these previous notions so that we can have a feel of what’s going on.
Notice that the two last sets have no maximal nor minimal element.
As a work out the reader can try to find , , , and of the empty set.
And now we’ll state the Completeness Axiom and with it our basic study of the real numbers will be complete.
Axiom 8 (Completeness Axiom) Any non-empty subset of with an upper bound has a real supremum. |
Theorem 9 Any non-empty subset of which is bounded from below has an infimum.
Proof: Omitted. |
Theorem 10 isn’t bounded above.
Proof: Omitted. |
Theorem 11 (Archimedean property)
Proof: Omitted. |
Theorem 11 tell us that in infinitesimals don’t exist: if we add any given quantity ( it doesn’t matter how small it is ) a sufficient number of times the value of the sum will always be greater than any other given quantity.
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