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]

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- Oct 22nd 2009, 05:53 PMamm345Proof-- Limits (Missed Class, please help)
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 PMartvandalay11
- Oct 22nd 2009, 06:21 PMamm345
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 PMartvandalay11

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