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Homework Help: Check my three proofs

  1. Jul 31, 2012 #1
    I am doing exercises form Velleman's How to Prove It

    1. The problem statement, all variables and given/known data

    1. Suppose a and b are real numbers. Prove that if a < b < 0 then a^2 > b^2.
    2. Suppose a and b are real numbers. Prove that if 0 < a < b then 1/b < 1/a.
    3. Suppose a and b are real numbers. Prove that if a < b then (a+b)/2 < b.

    2. The attempt at a solution

    1. Since a < b < 0 , this means that both a and b are negative numbers. Then i multiply
    a < b by a and get a^2 > ab and then again multiply a < b by b and get ab > b^2.
    Then since, a^2 > ab and ab > b^2 therefore a^2 > b^2.

    2. I multiply a < b by ab and get 1/b < 1/a.

    3. I add b to a < b and get a + b < 2b and divide this by 2 and get (a+b)/2 < b
  2. jcsd
  3. Jul 31, 2012 #2


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    Science Advisor

    Yes, that's excellent.

    Well, you meandivide by ab, not multiply. And you should specifically say that "because 0< a< b, ab> 0".

    Yes. Notice that the same kind of argument shows that a< (a+b)/2 so that the "mean" of two distinct numbers always lies between them.
  4. Jul 31, 2012 #3
    Thank you for response.

    2. Yes i meant divide not multiply.
    3. I haven't noticed that but it's a nice little insight.

    So, is the key in proofs to manipulate equation to get for A to B without putting in numbers since that isn't proof?
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