4

color=#000 Let $a_1,a_2,...,a_n$ be n real numbers. Show that there exists some real number $\alpha$ such that $a_1+\alpha,a_2+\alpha,...,a_n+\alpha$ are all irrational. \\ color=#000 Prove that such a satetement is not valid if all these are rquired to be rational. \\ Hint (given in question) : Use Pigeon Hole Principle