Guide To Walter Rudin's Principles, 1.17, 1.18 (Proof Details)

1.17: The definitions of an ordered field $F$, for $x,y,z \in F$
[i] $x+y \lt x+z$ if $y\lt z$
[ii] $xy \gt 0$ if $x\gt 0 \land y \gt 0$
Review the basic rules of working with inequalities.

1.18.a. Prove
$x \gt 0 \implies -x \lt 0$
$$ \begin{array}{l c} x \gt 0 & \\ -x + x \gt -x + 0 & \\ 0 \gt -x & \end{array} $$ 1.18.b. Prove
$x\gt 0 \land y \lt z \implies xy \lt xz$ $$ \begin{array}{l c} y \lt z & \\ y-y \lt z-y & \\ 0 \lt z-y & \\ 0x \lt (z-y)x & (?\times x) \ since \ x \gt 0 \\ 0 \lt zx - yx & \\ yx \lt zx -yx + yx & \\ yx \lt zx \end{array} $$ 1.18.c. Prove
$x\lt 0 \land y \lt z \implies xy \gt xz$ $$ \begin{array}{l c} y \lt z & \\ y-y \lt z-y & \\ 0 \lt z-y & \\ 0x \lt -(x(z-y)) & \ since \ x \lt 0 \\ 0x \lt (-x)(z-y) & 1.16.c \\ 0 \lt (-x)z + (-x)(-y) & \\ 0 \lt -xz + (-x)(-y) & \\ 0 \lt -xz + (-(-xy)) & \\ 0 \lt -xz + xy & \\ xz \lt xy \end{array} $$ 1.18.d. Prove
$x \not = 0 \implies x^2 \gt 0$
If $x\gt 0$ then $xx \gt 0$ by 1.17[ii].
If $x\lt 0$ then $$ \begin{array}{l c} (-x) \gt 0 & \\ (-x)(-x) \gt 0 & 1.17(ii) \\ xx \gt 0 & 1.16.d \end{array} $$ 1.18.e. Prove
$0 \lt x \lt y \implies 0 \lt y^{-1} \lt x^{-1} $
If $y\gt 0 \land v \leq 0$ then
$$ \begin{array}{l c} yv \leq 0 & try \ v = y^{-1}\\ y \times y^{-1} = 1 \gt 0 & false \ so \ v \ (y^{-1}) \gt 0\\ \hline x \lt y & \\ (x^{-1}y^{-1})x \lt (x^{-1}y^{-1})y & \\ y^{-1} \lt x^{-1} \end{array} $$

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