Have a question:
Suppose a, b can be any real number and for every e greater than 0, a is less than or equal to b +e. Prove that a is less than or equal to b.
It makes sense, just thinking about it, but how do you actually prove it?
MK
Let e = a - b > 0 (the last step is by assumption of a > b.)
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?
-Dan