The irrational numbers are dense
Task number: 2830
Show that the set of irrational numbers is dense in the set of the real numbers, i.e. that in every non-empty open interval it is possible to find an irrational number. You may use the fact that the set of rational numbers is dense in the reals.
It suffices to solve the exercise in a bounded interval \((a,b)\), because every unbounded interval contains a bounded subinterval. Using the density of the rational numbers, we know that there is some rational number \(q\) in the interval \((a - \sqrt 2, b - \sqrt 2)\). Then \(q + \sqrt 2\) is an irrational number in the interval \((a, b)\).