p - adic numbers , LTCC 2010

The following is a proof which is independent of this characterisation. First assume that ‖ ‖ is non-archimedean. Let x, y ∈ K. Using that ‖ ‖ extends | | we then obtain |x + y| = ‖x + y‖ ≤ max{‖x‖, ‖y‖} = max{|x|, |y|} which shows that | | is non-archimedean. Now assume that | | is non-archimedean. Let x, y ∈ K̂. Let ε > 0. Since K is dense in K̂ there exist u, v ∈ K such that ‖x − u‖ < ε and ‖y − v‖ < ε. Hence it follows that (using that ‖ ‖ extends | | and that ‖ ‖ is an absolute value) |u| = ‖u‖ = ‖u− x + x‖ ≤ ‖u− x‖+ ‖x‖ < ε + ‖x‖. Similarly |v| < ε + ‖y‖. Therefore ‖u + v‖ = |u + v| ≤ max{|u|, |v|} < max{ε + ‖x‖, ε + ‖y‖} = ε + max{‖x‖, ‖y‖}.