Problem 15. Prove the following statements. Once you have proven the statement, if needed, feel free to use that in the subsequent statement by giving a reference to it (e.g. it follows from Lemma 1 or from part(a) it follows that...). a+b 2 (a). Let a, b ER such that a < b. Show that their average s = will lie between a and b. (b). Suppose r is a rational number and a is an irrational number. Prove that r + a must be an irrational number. (c). Prove that between every rational and an irrational number we can find an irrational number that lies between them.