Proposition: Product of Convegent Real Sequencesdeleteeditadd to favorites[id:1135]vote: last edited 2 months ago by bookofproofs Let \((a_n)_{n\in\mathbb N}\) and \((b_n)_{n\in\mathbb N}\) be convergent real sequences to the limits \(\lim_{n\rightarrow\infty} a_n=a\) and \(\lim_{n\rightarrow\infty} b_n=b\). Consider the real sequence \((c_n)_{n\in\mathbb N}\) with \(c_n:=a_n \cdot b_n\). Then \((c_n)_{n\in\mathbb N}\) is also convergent and its limit equals \(\lim_{n\rightarrow\infty} c_n=a \cdot b\). This proposition can be expressed in the short form: \[\lim_{n\rightarrow\infty} (a_n \cdot b_n)=\lim_{n\rightarrow\infty} a_n \cdot \lim_{n\rightarrow\infty} b_n.\] References [581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983 All logical predecessors and all successors of the current node are: Found a mistake? Fix it by adding/editing the corresponding proof(s) below!Learn more about the axiomatic method. Subordinated Structure: Contribute to BoP: add a new Proof add add a new Axiom add add a new Definition add add a new Motivation add add a new Example add add a new Application add add a new Explanation add add a new Interpretation add add a new Corollary add add a new Proposition add add a new Lemma add add a new Theorem add add a new Algorithm add add a new Open Problem add add a new Bibliography (Branch) add add a new Comment (Branch) add |
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