# Homework Help: Real Analysis Intervals Proof

1. Sep 19, 2009

### cmajor47

1. The problem statement, all variables and given/known data
Prove that I<J if and only if x<y for every x$$\in$$I and y$$\in$$J.

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

3. 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
Therefore the original statement is true, I<J if and only x<y for every x$$\in$$I and y$$\in$$J.