Show that the irrational numbers are dense in R. [Hint: Use the fact that sqrt(2) is irrational.]
Book already proved that the rationals are dense in R.
The Attempt at a Solution
My intuition is that I'm supposed to make some sort of argument that since the rationals are dense in R, then just by transposing the rationals by sqrt(2), I'm showing that irrationals are also dense. (Using that a rational plus an irrational is an irrational.)
I just can't figure out for the life of me how to start this argument.
I could just say
x < r < y ->
x + sqrt(2) < r + sqrt(2) < y + sqrt(2) ->
a = x + sqrt(2), b = y + sqrt(2) ->
a < r + sqrt(2) < b
But that feels horribly unprooflike...and just quite horrible. I want something better than that. I also feel like this should be horribly easy, but I just can't seem to get anything worthwhile.
Give me as little information as you can please. If anything, I just need a point in the right direction, just a suggestion of how I should be venturing to prove this.
Thanks for your help.