1. ## 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.

(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.

2. ## 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.