Proving InfE <= SupE, Proving 1/n < a < n, and Proving N = E

[mollysmiith]

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 ! :)

 

[╔(σ_σ)╝]

Please show us your attempt at a solution.

 

[mollysmiith]

i can not some up with anything i am not sure where to start for any of them

 

[╔(σ_σ)╝]

Do you know the definitions of inf A and sup A ?