1. ## Basic Proof

Let $\displaystyle x$ and $\displaystyle y$ be distinct real numbers. Prove that there is a neighborhood $\displaystyle P$ of $\displaystyle x$ and a neighborhood $\displaystyle Q$ of $\displaystyle y$ such that $\displaystyle P \cap Q = \emptyset$.

The book left this proof as an exercise and the instructor said nothing about it. I'm just curious as to what the proof might be.

2. ## Re: Basic Proof

Have you considered a neighborhood smaller than ½|P-Q|?

3. ## Re: Basic Proof

I have tried to consider your suggestion, and it looks similar to the epsilon we chose when proving that a sequence can't converge to two numbers unless they were equal. However, I'm not quite sure what I'm supposed to do with that... We just assumed the OP was true and moved on with it.

P.S. I don't have much experience proving things so it's taking me awhile to think things through.

4. ## Re: Basic Proof

Originally Posted by Aryth
I have tried to consider your suggestion, and it looks similar to the epsilon we chose when proving that a sequence can't converge to two numbers unless they were equal. However, I'm not quite sure what I'm supposed to do with that... We just assumed the OP was true and moved on with it.
P.S. I don't have much experience proving things so it's taking me awhile to think things through.
Let $\displaystyle \delta=\frac{|x-y|}{4}>0$.
Let $\displaystyle P=(x-\delta,x+\delta)~\&~P=(y-\delta,y+\delta)$.
Show that $\displaystyle P\cap Q=\emptyset$.

5. ## Re: Basic Proof

Just curious, why did you pick that as $\displaystyle \delta$?

6. ## Re: Basic Proof

Originally Posted by Aryth
Just curious, why did you pick that as $\displaystyle \delta$?
Because it works.

7. ## Re: Basic Proof

That's basically the same thing my instructor said. To me it just seems random, but it does work.

8. ## Re: Basic Proof

The point is that if a specific $\displaystyle \delta$ works, then so does any positive number less than $\displaystyle \delta$.

TKHunny suggested $\displaystyle \deltas= \frac{|P- Q|}{2}$ which is the largest number that would work so that $\displaystyle N_P(\delta)$ and $\displaystyle N_Q(\delta)$ are disjoint because $\displaystyle \delta+ \delta= \frac{|P-Q|}{2}+ \frac{|P-Q|}{2}= |P- Q|$. (Draw a picture to see why.) Plato just chose to pick a smaller number. $\displaystyle \delta= \frac{|P-Q|}{3}$ would also work.

And, that is using the same $\displaystyle \delta$ for both sets, which is not necessary. You could also use, say, $\displaystyle \frac{|P-Q|}{5}$ for one open set and $\displaystyle \frac{|P-Q|}{3}$ for the other- just use two numbers that add up to less than |P-Q|.

9. ## Re: Basic Proof

Ah, that makes it clearer. Thank you very much.