State and prove Archimedean Property in IR

 State:  N is not bounded above in R. That is, given any x ∈ R, there exists n ∈ N such that x > n.

Proof: We prove this result by contradiction. Assume that N is bounded above in R. By the LUB property of R, there exists α ∈ R such that α = lub N. Then for each k ∈ N, we have k ≤ α. Since we wish to exploit the fact that α is the LUB of N and since we are dealing with integers, we consider α − 1 < α. Then α − 1 is not an upper bound of N and hence there exists N ∈ N such that N > α − 1. Adding 1 to both sides yields N + 1 > α. Since N + 1 ∈ N, we are forced to conclude that α is not an upper bound of N. Hence our assumption that N is bounded above is wrong.