Infinite amount of (ir)rationals between any two reals

[irlaxgod]

I need to prove by math induction, that there are infinite amout of rationals(also irrationals after but these two proofs come hand in hand) between x,y (where x and y are real) where x<y. i can prove there is one rational( or irrational) x<r<y where r =m/n (m,n are naturals numbers) . I just don't know how to set up for induction, i feel like i'm stupid cause i know its true just don't know how to prove it .

 

[pka]

I need to prove by math induction, that there are infinite amout of rationals(also irrationals after but these two proofs come hand in hand) between x,y (where x and y are real) where x<y. i can prove there is one rational( or irrational) x<r<y where r =m/n (m,n are naturals numbers) . I just don't know how to set up for induction, i feel like i'm stupid cause i know its true just don't know how to prove it .

Then there is a rational r[sub:3kjxg3wm]1[/sub:3kjxg3wm] between x and r.
Then there is a rational r[sub:3kjxg3wm]2[/sub:3kjxg3wm] between x and r[sub:3kjxg3wm]1[/sub:3kjxg3wm].
...
Then there is a rational r[sub:3kjxg3wm]k+1[/sub:3kjxg3wm] between x and r[sub:3kjxg3wm]k[/sub:3kjxg3wm].