implication of a set of inequalities

Per · #1 $Old$ November 18th, 2009, 10:09 AM

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:

INEQUALITY 1:

X – Y ≥ ε λ

INEQUALITY 2:

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!!

tonio · #2 $Old$ November 18th, 2009, 11:13 AM

Quote:

Originally Posted by Per $View Post$

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:

INEQUALITY 1:

X – Y ≥ ε λ

INEQUALITY 2:

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.

Not at all! To show something general is not true it is enough ONE single counterexample...for example, $X=2\,,\,Y=1\,,\,\epsilon=3\,,\,\lambda=45$

Tonio

I would be very grateful for all help I can get!!

.