Ordering

Theorem 215 (Landau 121). If ξ > η then η < ξ.

Each says that some upper ratio for η is a lower ratio for ξ.

Theorem 216 (Landau 122). If ξ < η then η > ξ.

Each says that some lower ratio for η is an upper ratio for ξ.

Theorem 217 (Landau 123). For any ξ and η, exactly one of ξ = η, ξ > η, ξ < η is the case.

Proof: 1) At most one. ξ = η excludes each of the others by Definitions 44 to 46. If ξ > η and ξ < η both held, some lower ratio X for ξ would be upper for η, and some upper ratio Y for ξ would be lower for η. By 2) of Definition 43 applied to ξ, X < Y; applied to η, Y < X. 2) At least one. If ξ ≠ η, some ratio is lower for one and not for the other. If it is lower for ξ and upper for η, then ξ > η. If lower for η and upper for ξ, then ξ < η.

Part 2) rests on its being determined, of each ratio, whether it is lower or upper for each bound. It does not require that we be able to find out.

Theorem 218 (Landau 124). If ξ ≧ η then η ≦ ξ. Theorem 215.

Theorem 219 (Landau 125). If ξ ≦ η then η ≧ ξ. Theorem 216.

Theorem 220 (Landau 126). If ξ < η and η < ζ, then ξ < ζ. (Transitivity of Ordering)

Some upper ratio X for ξ is lower for η, and some upper ratio Y for η is lower for ζ. By 2) of Definition 43 applied to η, X < Y. So Y is upper for ξ by Theorem 213, and it is lower for ζ.

Theorem 221 (Landau 127). If ξ ≦ η and η < ζ, or ξ < η and η ≦ ζ, then ξ < ζ.

Obvious if an equality holds; otherwise Theorem 220.

Theorem 222 (Landau 128). If ξ ≦ η and η ≦ ζ, then ξ ≦ ζ.

Obvious if two equalities hold; otherwise Theorem 221.