Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Mathematics
Calculus
Proof of convergence & divergence of increasing sequence
Reply to thread
Message
[QUOTE="hikarusteinitz, post: 5581715, member: 605065"] I'm using the book of Jerome Keisler: Elementary calculus an infinitesimal approach. I have trouble understanding the proof of the following theorem. I'm not sure what it means. [B]Theorem[/B]:[I] "An increasing sequence <S[SUB]n[/SUB]> either converges or diverges to infinity."[/I] [B]Proof:[/B] Let T be the set of all real numbers x such that x≤S[SUB]n[/SUB] for some n. [U]Case 1[/U]: T is the whole real line. If H is infinite we have x≤S[SUB]H[/SUB] for all real numbers x. So S[SUB]H[/SUB] is positive infinite and <S[SUB]n[/SUB]> diverges to ∞. [U]Case 2:[/U] T is not the whole real line. By the completeness theorem, T is an interval (-∞,b] or (-∞,b). For each real x<b, we have : x≤S[SUB]n[/SUB]≤S[SUB]n+1[/SUB]≤S[SUB]n+2[/SUB] . . . ≤b for some n. It follows that for infinite H, S[SUB]H[/SUB]≤b and S[SUB]H[/SUB]≈b. therefore, S[SUB]H[/SUB] converges to b. The book states the definition of an interval as the completeness axiom: [B]Completeness Axiom:[/B] [I]"Let A be a set of real numbers such that whenever x and y are in A, then any real number between x and y are in A. Then A is an Interval."[/I][U]Questions:[/U] 1.) When it says "Let T be the set of all real numbers x such that x≤S[SUB]n[/SUB] for some n". [U]What does it mean? [/U]"some n" means not just one n but maybe a few ns. [U]Or does it mean that as long as x is less than some some element of the sequence Sn then it s part of the set T?[/U] English isn't my first language. 2.) If x≤S[SUB]k[/SUB], then x≤S[SUB]k[/SUB]≤S[SUB]k+1[/SUB]≤S[SUB]k+2[/SUB] . . . because <S[SUB]n[/SUB]> is increasing. Then the set T must include all x≤S[SUB]H[/SUB] where H is infinity. [U]Did I understand it correctly?[/U] Again I think it means that as long as x is less than some some element of the sequence Sn then it is part of the set T. 3.) I think I understand case 1, but please check if I really understood it. My understanding is that: Since T is the whole real line then x can be any real number and since x≤S[SUB]n[/SUB] for some n, then x≤S[SUB]n[/SUB]≤S[SUB]n+1[/SUB]≤S[SUB]n+2[/SUB] . . .S[SUB]∞[/SUB]. Then x≤S[SUB]∞[/SUB] where x is any real number you may think of. S[SUB]∞[/SUB] is positive infinite. 4.)In case 2. If T is not the whole real line, it's easy to visualize why it is an interval (-∞,b] or (-∞,b),but I don't see how it follows from the completeness axiom. It might require several logical steps, but it does not follow immediately, at least for me. [U]But if T is (-∞,b] or (-∞,b), why x<b only why not x≤b?.[/U] The rest is just a bit hazy for me, I get it a bit but not clear enough. [U]Please explain case 2[/U]. Thanks in advance. [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
Calculus
Proof of convergence & divergence of increasing sequence
Back
Top