Introduction to Mathematical Analysis I - 3rd Edition

31 1.6.3 Let x ∈R. Prove that for everyn∈N, there is r ∈Qsuch that |x−r| < 1 n . 1.6.4 Prove that if x is a rational number andy is an irrational number, thenx+y is irrational. What can you say about xy? 1.6.5 Prove that in between two real numbers x and y with x <y, there are infinitely many rational numbers. 1.6.6 Prove that in between two real numbers x andy withx<y, there are infinitely many irrational numbers.

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