 # 1. Use only the field and order axioms to prove the following state...

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1. Use only the field and order axioms to prove the following statements for .2, y, Z € R. Make sure you indicate which axiom you are utilizing at each step. (a) Prove that if r > 0 and y > 0, then x + > 0. (b) Prove that if r > 0 and y > 0, then xy > 0. (c) Prove that for z + 0 and y + O,then (d) Prove that 0 is its own additive inverse, i.e. 0 - -0. (c) Prove that 1 is its own multiplicative inverse, i.e. 1 (f) Prove that r - y - 0 iff r - y. (g) Prove that if xy - Z and any two of x,y and Z are positive, then the third real number must also be positive. (h) Prove that if . > 0 then also =-1 > 0. 2. Given that [ - { -r, x, if 0 if x 0 prove the following statements for z, y € IR : (a) 0 for all I € IR: and - 0 iff 7 - 0. (b) [ - 1-z% (c) - I < (d) S y - -y S Z y. (e) x - y . (f) ||x| - [y| 1s/2+y/. 3. For the following sets A, find (if they exist) max A, min A, (sup A, inf (A) : (a) A - {1,3,9,4,0} . (b) A - [0,00). (c) A - [-1,3] - (d) A - {x:-2 - 1 - -0}. (e) A - {x R: I2 < 2}. (f) A - (-00,00) . (g) A - } -

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