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Math Help - Help with proving and implying inequalities please

  1. #1
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    Help with proving and implying inequalities please

    Hi,

    I've recently started a unit on algebra, but it's been a while since I finished high school and I haven't used any of this stuff since then so I'm struggling to remember what I'm supposed to do.

    The question reads:

    (i) Prove that a > 1 implies a^2 > 1.
    (ii) Prove that a^2 > 1 does not imply a > 1 by giving a counterexample.
    (Hint: You may use the following property of positive numbers a, b, c: If a < b then a + c < b + c and ca < cb. )

    I think I've done part (i) correctly by saying:
    a>1
    a.a>1.a
    a^2>a>1
    therefore, a>1 implies a^2>1

    However I'm getting really confused with part (ii). I'm assuming my last line needs to read something along the lines of a^2>1>a, but I just can't wrap my head around how b and/or c fit in and what formula I need to show to get to that point.

    Any help would be greatly appreciated.
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  2. #2
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    Re: Help with proving and implying inequalities please

    (i)
    \displaystyle \begin{align*} a &> 1 \\ a^2 &> 1^2 \\ a^2 &> 1 \end{align*}

    (ii)
    \displaystyle \begin{align*} a^2 &> 1 \\ \sqrt{a^2} &> \sqrt{1} \\ |a| &> 1 \\ a < -1 \textrm{ or } a &> 1 \end{align*}

    So any number \displaystyle \begin{align*} < -1 \end{align*} will serve as a conterexample.
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