Definition 43 (Landau 28). A bound is given when, of any ratio, it is determined whether it is a lower ratio or an upper ratio for that bound, in such a way that
- some ratio is lower and some ratio is upper;
- every lower ratio is less than every upper ratio;
- for every lower ratio there is a greater lower ratio.
The bound is then the relative maximum of its lower ratios and the relative minimum of its upper ratios. Small Greek letters stand for bounds.
The definition speaks of any ratio, one at a time. It never takes the ratios together as one whole. Landau's set of rational numbers is not used.
Definition 44 (Landau 29). ξ = η when every lower ratio for ξ is a lower ratio for η and every lower ratio for η is a lower ratio for ξ. Otherwise ξ ≠ η.
Theorem 212 (Landau 116 to 118). ξ = ξ. If ξ = η then η = ξ. If ξ = η and η = ζ then ξ = ζ.
These follow from the meaning of Definition 44.
Theorem 213 (Landau 119). If X is an upper ratio for ξ and X1 > X, then X1 is an upper ratio for ξ.
If X1 were lower it would be less than X, by 2) of Definition 43.
Theorem 214 (Landau 120). If X is a lower ratio for ξ and X1 < X, then X1 is a lower ratio for ξ.
If X1 were upper it would be greater than X, by 2) of Definition 43.
Remark. Conversely, Theorem 214 is equivalent to 2). So to show that a sorting of ratios gives a bound, it is enough to show: 1) some ratio is lower and some is not; 2) with any lower ratio, every lesser ratio is lower; 3) with any lower ratio, some greater ratio is lower.