If and for all , then .

* Proof. * Suppose otherwise, that

for all

, but

. Then, we have

and

implies

, so

is a positive real number. Then let

(since

this is a valid choice of

). By assumption we then have

(since

for all

), but this is a contradiction since

(equality is symmetric) implies

. Therefore, we must have

*Related*

Even when is well known that “equality” is an equivalence relation (i.e symmetric) we can prove that x<x cannot hold since if it happens, then x-x is in R+, which means, 0 is in R+, leading to a contradiction with the first Axiom of order.

Can you please explain how you are taking h=x ? I am not able to understand it.