Intervals with Natural Number Endpoints

[Bashyboy]

Homework Statement

Show that $(n,n+1) \cap (k,k+1)$ is empty, provided that $n \neq k$.

Homework Equations

The Attempt at a Solution

[/B]
WLOG, take $k < n$. Then $k -n \ge 1$ is some natural number. If $x \in (n,n+1) \cap (k,k+1)$, then $-(n+1) < -x < -n$ and $k < x < k+1$. Adding the two inequalities together, we obtain $k-n-1 < 0 < k-n + 1$ or $0 < k-n < 1 < k-n + 2$, which contradicts the fact that $k-n$ is some natural number.

How does this sound? Any better alternatives?

 

[FactChecker]

WLOG, take $k < n$. Then $k -n \ge 1$ is some natural number.

That is wrong. k-n < 0

 

[Bashyboy]

That is wrong. k-n < 0

Whoops! I meant to say $n < k$. That should fix everything.

 

[FactChecker]

I think you can prove it more directly. I don't follow the last part, but it might be right.