Proving the Inductive Property of Natural Numbers - Can You Help?

[mollysmiith]

Analysis math help please!?
1. Let E be a nonempty subset of R (real numbers)

Prove that infE <= supE

2. Prove that if a > 0 then there exists n element N (natural) such that 1/n < a < n

3. A subset E of te real numbers R is an inductive set if

i) 1 element E
ii) If x element E then x + 1 element E

A real number is called a natural number if it belongs to every inductive set. The set of natural numbers is denoted by N. Recall that the principle of mathematical inductions says that if M is any subset of N that is an inductive set then M = N. Show that N = E, where E = {1,2,3,4...}

Any help would be greatly appreciated ! :)

 

[HallsofIvy]

Analysis math help please!?
1. Let E be a nonempty subset of R (real numbers)

Prove that infE <= supE

The infimum of E is defined as a lower bound for E: inf E<= x for any x in E.
The supremum of E is defined as an upper bound for E: x<= sup E for any x in E.
Put those two together. Do you see why the fact that E is non-empty is important?

2. Prove that if a > 0 then there exists n element N (natural) such that 1/n < a < n

I don't know what theorems you have to base this on but one fundamental property is the "Archimedean property": given any real number, a, there exist an integer such that a<n. Of course, if a> 1, then 1< n also so that 1/n< 1< a. If 0< a< 1, look instead at 1< 1/a< n. In that case, 1/n< a and, of course, a< 1< n.

3. A subset E of te real numbers R is an inductive set if

i) 1 element E
ii) If x element E then x + 1 element E

A real number is called a natural number if it belongs to every inductive set. The set of natural numbers is denoted by N. Recall that the principle of mathematical inductions says that if M is any subset of N that is an inductive set then M = N. Show that N = E, where E = {1,2,3,4...}

Show that every inductive set contains, at least, {1, 2, 3, 4, ...}.

Any help would be greatly appreciated ! :)

Note that the definition "inductive set" above does not restrict its members to the natural numbers. For example, the set {1/2, 1, 3/2, 2, 5/2, 4, 9/2, 5...} is an "inductive set". Another is {2/3, 1, 5/3, 2, 8/3, 3, 11/3, 4, ...}. But you can prove that N is a subset of every inductive set.