• Oct 22nd 2009, 05:53 PM
amm345
Let
x, y be elements ofR such that x not= y. Prove that there exists epsom > 0 such that y not element of[x − epsom, x + epsom]

• Oct 22nd 2009, 06:03 PM
artvandalay11
Quote:

Originally Posted by amm345
Let
x, y be elements ofR such that x not= y. Prove that there exists epsom > 0 such that y not element of[x − epsom, x + epsom]

Since $\displaystyle x\not = y$, distance between x and y, denoted $\displaystyle d(x,y)\not = 0$

Define $\displaystyle \epsilon=d(x,y)/2$

The distance function is never negative so epsilon is greater than 0

Then....

can you finish it from here?
• Oct 22nd 2009, 06:21 PM
amm345
I'm not sure, I had kinda had that based on the lecture notes he posts online, but I'm still not sure what to do next.
• Oct 22nd 2009, 06:48 PM
artvandalay11
Quote:

Originally Posted by artvandalay11
Since $\displaystyle x\not = y$, distance between x and y, denoted $\displaystyle d(x,y)\not = 0$

Define $\displaystyle \epsilon=d(x,y)/2$

The distance function is never negative so epsilon is greater than 0

Then....

can you finish it from here?

Assume $\displaystyle y\in[x-\frac{d(x,y)}{2}, x+\frac{d(x,y)}{2}]$

Then $\displaystyle x-\frac{d(x,y)}{2}\leq y\leq x+\frac{d(x,y)}{2}$

So $\displaystyle -\frac{d(x,y)}{2}\leq y-x\leq\frac{d(x,y)}{2}$

On the real line, $\displaystyle d(x,y)=|x-y|$ so

$\displaystyle -\frac{1}{2}|x-y|\leq y-x\leq \frac{1}{2}|x-y|$

So $\displaystyle -\frac{1}{2}|x-y|\leq y-x$ and

$\displaystyle y-x\leq\frac{1}{2}|x-y|$

Multiplying by -2 and 2 respectively yields

$\displaystyle -2(y-x)\geq |x-y|$ and

$\displaystyle 2(y-x)\geq |x-y|$

These two facts imply x=y, but that is a contradiction with the given information