Solution of the 4th Homework
نویسنده
چکیده
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 x n ≥ N . Then we get k ≥ N ⇒ |an − ak | < ε′. In view of Proposition 6.30, taking formal limit LIMk→∞, we obtain |an − x | ≤ ε′ ⇒ |an − x | < ε. In summary, for any positive real ε > 0 we was able to nd N ∈ N such that |an − x | < ε whenever n ≥ N . This proves the desired statement at the beginning of the proof and hence an converges to x in R. ////
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