Proof of Observation 1. Before the proof, we remind that what we want to prove is the following statement: (∀ε ∈ R) ε > 0⇒ ((∃N ∈ N)(∀n ∈ N) n ≥ N ⇒ |an − x | < ε). To this end, choose arbitrary positive real ε > 0 and pick a positive rational ε′ with 0 < ε′ < ε by utilizing the Archimedean property. Then there exists N (depending on ε′ and hence on ε) such that j, k ≥ N ⇒ |a j − ak | < ε′. Now...