Problem 15. Prove the following statements. Once you have proven the statement, if needed, feel free...

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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. 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.

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