Prove sqrt(2) in R by showing x · x = 2 where x = A|B is the cut in Q with A = {r in Q : r ≤ 0 or r² < 2}

Question: Prove by showing where is the cut in with .

Let be Dedekind cut with

Note (this can be proved separately for cut ). Since , , . We show .

Case . Let . Since , by Archimedes principle, exists s.t. . Then, By density of , exists s.t. . So and thus . Since and , contradiction that is upper bound of .

Case . Let . Since , for some . By density of , exists s.t. so and hence for all and is thus upper bound smaller than supposed lowest upper bound , contradiction.

Hence so .