Read through the following theorem and proof and answer the questions on it that follow. Be sure to number your responses clearly and correctly. Theorem: Let F be an ordered field, and let a, b e F with a, b > 0. If a² < b² then a < b. Proof: (1) Suppose that a? < b², i.e. a² – b2 < 0 where a, b > 0. (2) This means that (a + b)(a – b) < 0. (3) From this we get that (a + b)-1. [(a+ b)(a – b)] < (a + b)-1 . 0. (4) Hence a – b < 0, i.e. a < b. QED - 1.1 Explain why the inverse (a + b)-1 of a + b exists. 1.2 Name the three field properties needed that one makes use of in reducing the expression (a + b)-1 . [(a + b)(a – b)] on the LHS of (3) to a – 6.

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