Theorem:If R is an ordered field, then for any positive element x there shall exist another positive element y such that y<x.

Proof of Problem:

Assume by contradiction that a>b.

Then, a=b+c where c is positive, i.e. c=b-a.

Choose b1=b+d where 0<d<c as in above theorem.

Then,

b1=b+d<b+c=a and b+d=b1>b.

A contradiction.

Thus, a<=b.

Q.E.D.