I'll write theorem first.
Theorem: Every Archimedean ordered field can be order-embedded into R.
Proof: Let (K, ≤) be Archimedean. Then Q is dense in K. We define φ : K → R as follows. For a ∈ K, let: Ia = { r ∈ Q | r < a } and Ja = { s ∈ Q | s > a }.
There is some x ∈ R such that Ia ≤ x ≤ Ja, since R is Dedekind complete; moreover, there is only one such x, by the density of Q in R. We may therefore define φ(a) to be x.
For all a, b ∈ K,
φ(a + b) = φ(a) + φ(b),
φ(ab) = φ(a)φ(b)
ex. 1. Show that φ is a field embedding.
ex. 2. Show that φ preserves order.
Can someone tell me how can I simple show it?
Thx for help. Have a good day!