Is This Proof of Interval Order Correct?

[cmajor47]

Homework Statement

Prove that I<J if and only if x<y for every x$$\in$$I and y$$\in$$J.

Homework Equations

I=[r,s] and J=[u,v]
I<J means that s<u.

The Attempt at a Solution

Proof by contradiction: Assume that I<J if x>y for every x$$\in$$I and y$$\in$$J.
Let I be the interval [r,s] and J be the interval [u,v].
x$$\in$$I means r$$\leq$$x$$\leq$$s
y$$\in$$J means u$$\leq$$y$$\leq$$v
Since x>y, u$$\leq$$y<x$$\leq$$s
Therefore u<s
However the definiton of I<J is s<u
This contradicts our statement
Therefore the original statement is true, I<J if and only x<y for every x$$\in$$I and y$$\in$$J.

I just wanted to make sure that this proof is correct and complete.

 

[mighty2000]

Thank you!

Yes, your proof is correct and complete. You have correctly used the definition of I<J and the properties of intervals to show that the assumption leads to a contradiction, thus proving the original statement. Well done!