# Thread: Irrational number proof

1. ## Irrational number proof

I'm having trouble understanding how to prove this problem. I need some help.

Prove that for two arbitrary real numbers a and b, with a < b there is an irrational number c such that a < c < b. Hint(consider a/sqrt(2) and b/sqrt(2))

Thanks for any help

2. Originally Posted by tuyt6444
I'm having trouble understanding how to prove this problem. I need some help.

Prove that for two arbitrary real numbers a and b, with a < b there is an irrational number c such that a < c < b. Hint(consider a/sqrt(2) and b/sqrt(2))

Thanks for any help
Hint 2: What inequality can you make with the numbers in the hint?

Hint 3: Apply the density of $\displaystyle \mathbb Q$ in $\displaystyle \mathbb R$ property and note that $\displaystyle r \sqrt 2$ is irrational if $\displaystyle r \in \mathbb Q$...which pretty much is telling you how to solve the problem.