Prove Proposition 0 x y and 0 u v ux vy You might use som

Prove Proposition: [0 x < y and 0 u < v] ux < vy

You might use some of the following definitions:

1 For every x R, exactly one of the following hold: x > 0, x = 0, or x > 0.

2 For all x, y in R we have: [x > 0 and y > 0] x + y > 0.

3 For all x, y in R we have: [x > 0 and y > 0] xy > 0.

Please indicate which definition have you used for each step. Thanks.

Solution

as given we need to prove that if 0 x < y and 0 u < v ux < vy

we have,

0 x < y

we can say that,

x < y

multiply with u on both the side we have,

ux < uy --------------------1)

we have,

0 u < v

we can say that,

u < v

multiply both side with y we have,

uy < vy ----------------2)

from equation 1) and 2) we have,

ux < uy and uy < vy we can say that,

ux < vy