I have had problems trying to prove that one inequality does not imply another inequality. Let ε, λ, X and Y be scalars where
ε ≥ 0
λ > 0
-∞ < X < +∞
-∞ < Y < +∞
We have the following two inequalities:
X – Y ≥ ε λ
if ε = 0 then X – Y ≥ 0
if ε > 0 then X – Y > 0
I want to show that INEQUALITY 1 implies INEQUALITY 2, but that the converse is not true. That is,
i) If INEQUALITY 1 is satisfied then INEQUALITY 2 must be satisfied
ii) If INEQUALITY 2 is satisfied then INEQUALITY 1 need not be satisfied
While i) is easy to prove, I have gotten stuck on trying to prove ii). The thing is that I need to show it in the general case, and not with a specific example using made up numbers.
I would be very grateful for all help I can get!!