$forall xinmathbb{R}enspace enspace, forall yinmathbb{R} : (xge y)$ or $(xle y)$ proof?

It comes from definition of Real numbers - Cauchy sequences, Dedekind cuts or other. In any way aim is to have so called The complete ordered field.

1. Dedekind cuts:

We take rational numbers $mathbb{Q}$ and consider its $alpha$ subsets with following properties: $$begin{array}{} alpha neq emptyset, alpha neq mathbb{Q} \ q in mathbb{Q} land p in alpha land q<p Rightarrow q in alpha \ p in alpha Rightarrow exists r in mathbb{Q}, p<r end{array}$$ Such subsets are called "cuts". We define for cuts "$alpha < beta$", when $alpha$ is proper subset of $beta$. Now can be obtained all desired properties of order, including least upper bound property.

2. Cauchy sequences:

Again we consider rational numbers $mathbb{Q}$ with metric $|x-y|$ and look at all Cauchy sequences. We called 2 Cauchy sequence equivalent, when they have same limit. By this we divide $mathbb{Q}$ into equivalence classes, which we called real numbers. Order for equivalence classes $(x_n) leqslant (y_n)$ iif $(x_n)$ is equivalent to $(y_n)$ or $exists N, n>N, x_n leqslant y_n$. Now again is possible to have all order properties.