Proving: Neighborhoods of x and y Have Empty Intersection

[Trung]

Homework Statement

Let x and y be real numbers. Prove there is a neighborhood P of x and a neighborhood Q of y such that P intersection Q is the empty set.

Homework Equations

The Attempt at a Solution

Sorry, I know this is elementary to many of you, but I am just starting out in this course and I need some hints on how to get started.

 

[EnumaElish]

It is asking "can you fit a pair of brackets around x and another pair of brackets around y such that the two pairs of brackets do not touch each other?" Remember, you are the one choosing how tight the first pair of brackets (around x) as well as how tight the second pair (around y). (You can make them as tight as you want.)

 

[Trung]

Pictorially it would be something like this:

<---------------(---x---)---------(-----y-----)--------------->

...but this diagram does not constitute a proof. I do not know how to make it into a rigorous argument. I have the sets (x-r, x+r) and (y-s, y+s) but I don't know what to do with them.

 

[HallsofIvy]

If x and y are different then they have some non-zero distance between them. Think about neighborhoods of x and y with radius equal to 1/3 that distance.