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.(adsbygoogle = window.adsbygoogle || []).push({});

Theorem:"An increasing sequence <S_{n}> either converges or diverges to infinity."

Proof:

Let T be the set of all real numbers x such that x≤S_{n}for some n.

Case 1: T is the whole real line. If H is infinite we have x≤S_{H}for all real numbers x. So S_{H}is positive infinite and <S_{n}> diverges to ∞.

Case 2: 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_{n}≤S_{n+1}≤S_{n+2}. . . ≤b

for some n. It follows that for infinite H, S_{H}≤b and S_{H}≈b. therefore, S_{H}converges to b.

The book states the definition of an interval as the completeness axiom:

Completeness Axiom:

"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."

Questions:

1.) When it says "Let T be the set of all real numbers x such that x≤S_{n}for some n". What does it mean? "some n" means not just one n but maybe a few ns. 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? English isn't my first language.

2.) If x≤S_{k}, then x≤S_{k}≤S_{k+1}≤S_{k+2}. . . because <S_{n}> is increasing. Then the set T must include all x≤S_{H}where H is infinity. Did I understand it correctly? 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_{n}for some n, then x≤S_{n}≤S_{n+1}≤S_{n+2}. . .S_{∞}. Then x≤S_{∞}where x is any real number you may think of. S_{∞}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. But if T is (-∞,b] or (-∞,b), why x<b only why not x≤b?. The rest is just a bit hazy for me, I get it a bit but not clear enough. Please explain case 2. Thanks in advance.

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# I Proof of convergence & divergence of increasing sequence

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