Simple proof in analysis (where did I go wrong)

Homework Statement

Let a,b be any real number. Show that if a-e < b for any e > 0, then a <= b.

The Attempt at a Solution

I tried a proof by contradiction, that if a>b then we violate some assumption.
Here it goes:

Suppose not.
Then a > b.
a- e > b - e
Since we assume a-e < b, thus b > b-e. But well...this seems right. Where did I go wrong?