Thread: Real Analysis Homework Help!!!!!!

a.)If y>0, then there exists n in N such that n-1(less than or equal to) y<n.
Prove



b.)Also: Prove that the n in part (a) is unique...


Any help would be greatly appreciated!!!!

Originally Posted by trojanlaxx223

a.)If y>0, then there exists n in N such that n-1(less than or equal to) y<n.
Prove



b.)Also: Prove that the n in part (a) is unique...


Any help would be greatly appreciated!!!!

Let A={nεN:n>y}.Now that set is not empty because the Archimedean principal tell us that:

for all y>0 there exists nεN SUCH THAT n>y

SINCE A is not empty and also a subset of the natural Nos N, then according to the well ordering principal of natural Nos :

there exists a kεA AND .

But since kεA ,THEN y<k. ALSO k-1<k.

If now y<k-1 ,then (k-1)εN which implies a contradiction ,hence :

For part (b).

Assume that the set A HAS two minimums k,m both belonging to A

But since m is a minimum , ...........(1)

Also since k a minimum , .............(2)

From (1) and(2) we conclude k=m ,therefor k is unique

Do not forget that a minimum of a set is an infemum that belongs to the set.

If you do not understand part (b) i can give a more detailed proof

Thanks for the help. This question was a part of my review sheet that I couldn't understand. The proof for uniqueness was perfect. Thanks again!!