# Proof of convergence & divergence of increasing sequence

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.