I am following Mathematical Analysis by Tom Apostol. I have a confusion here.
3. The half-open interval $S = [0,1)$ is bounded above by 1 but has no maximum element. It's minimum element is 0.
For these sets like in Example 3, which are bounded above but have no maximum element, there is a concept which takes place of the maximum element. It is called least upper bound or supremum as is defined as follows
And there is a property called Approximation property
Theorem (Approximation property): Let $S$ be a set of nonempty set of Real numbers with a supremum, say $b = \sup S$. Then for every $a < b$ there is some $x$ in $S$ such that $$a < x \leq b$$
My question is shouldn't it be like this $ a \leq x < b$ since the set $S$ does not contain b? Or does it imply that we approximate $b$ to be in $S$?
Also if the the maximum element of the set exist, then would it be the supremum?
And finally, there is a statement
... The completeness axiom allows us to introduce irrational numbers in real number system, and it gives the real number system a property of continuity that is fundamental to many theorems in analysis.
How does completeness axiom allows us to introduce irrational number in real number system?