Proving a<=b when a<=b1 for all b1>b in Real Analysis

[phygiks]

Hey guys, got stuck on this question while doing homework. I would appreciate any help.
Let a,b exist in reals. Show that if a<=b1 for every b1 > b. then a <= b.

I really got nowhere. I tried letting b1(n)=b+nE where E is a infinitesimal. Then a <= b+nE for all n. Don't really know how to use any axioms here either.

 

[Dick]

Do a proof by contradiction. Suppose it's NOT true that a<=b. Then a>b. Where is (a+b)/2? Use that for b1. You certainly don't need infinitesimals.

 

[phygiks]

I think I got it, so a>b, let b1 = (a+b)/2, then b1 > b, but a<= (b1=(a+b)/2). Contradiction

 

[Dick]

You've got it.