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. Theorem: "An increasing sequence <Sn> either converges or diverges to infinity." Proof: Let T be the set of all real numbers x such that x≤Sn for some n. Case 1: T is the whole real line. If H is infinite we have x≤SH for all real numbers x. So SH is positive infinite and <Sn> 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≤Sn≤Sn+1≤Sn+2 . . . ≤b for some n. It follows that for infinite H, SH≤b and SH≈b. therefore, SH 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≤Sn 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≤Sk, then x≤Sk≤Sk+1≤Sk+2 . . . because <Sn> is increasing. Then the set T must include all x≤SH 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≤Sn for some n, then x≤Sn≤Sn+1≤Sn+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.