Say it is false, then a>b.
Then a-b>0. So let e=a-b.
See what happens when you use the e in the given.
Thus b + e = b + (a - b) = a. (Which holds for the condition in the problem statement.)
Now, e is positive so there is some real number e' such that 0 < e' < e.
The problem statement says that a =< b + e' also, since any positive real number will do.
But e' < e implies b + e' < b + e.
Thus b + e' < a, contrary to the problem statement.
Thus a is not greater than b.
What can you do to show that a is not equal to b?