Toward the Bolzano Weierstrass theorem from scratch

Understand that mathematical systems are built on 'taken for granted' axioms (or accepted truths. These axioms are 'accepted' to be true without any proof. Any proof of any mathematical law is built on a set of axioms.

We will state one such axiom for real number system called the Completeness Axiom. From wikipedia we quote "The axiom states that every non-empty subset S of R that has an upper bound in R has a least upper bound, or supremum, in R"

Let us understand the different terms in the axiom or given truth:

1. R is bold = This symbol denotes the set (that is collection) of real numbers.
2. Real Number = collect all the fractions, integers, square-roots of non-square numbers and other irrational numbers like e, \(\pi\), and put them in one set. This makes the set of real numbers.
3. subset of R = collection of some (distinct) numbers from the set of real numbers.
4. upper bound = take the literal meaning
5. least upper bound = again take the literal meaning
6. supremum = another word for 'least upper bound'

Now we are set to proof a simple form of Bolzano - Weierstrass theorem. Understand that theorems are based on logical conclusions derived from axioms by reasoning. So we will use our Completeness Axiom to build the proof of the theorem.

Statement: If \(a__k\) is a monotone sequence of real numbers (e.g., if \(a__k ≤ a_{_{k+1}\}), then this sequence has a finite limit if and only if the sequence is bounded.

Please note that this is NOT the original statement of the Bolzano Weierstrass theorem. But for the purpose of high school mathematics we can use this clone. Stay tuned and we will continue