1. Hint: From the problem 1 above it follows that if r is rational then r + has to be irrational
(because we know that is). Now apply the theorem about the denseness of Q in R to the pair of
numbers a - and b - Given two numbers a, b R with a
Solution
If you proved the density of the rationals (between any two real numbers, there exists a rational
number), this is easy. Apply the density of Q to the interval (a - sqrt(2), b - sqrt(2)): There exists
r in Q such that a - sqrt(2) < r < b - sqrt(2). ==> a < r + sqrt(2) < b. Since the sum of a rational
and an irrational is irrational (which is easy to prove by contradiction, if you haven't proved this
yet), we are done.