Page 165 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

Proposition 7.2.3 (Real LUBGLB Proposition). The real numbers have both the least upper bound property
and the greatest lower bound property.

Proof: We need to show that every non-empty set which is bounded below has a greatest lower bound
in R . So, let M be a non-empty set which is bounded below. Let’s define

S = {s ∈ R such that − s ∈ M}.

Since M is bounded below, there is q ∈ Q such that for each m ∈ M ,

m ≥ q =⇒−m ≤−q.

This means that for each s ∈ S , since −s ∈ M ,

−s ≥ q =⇒ s ≤−q.

So −q is an upper bound for S . By the least upper bound property of the real numbers, S has a least
upper bound, x . Now notice that −− m = m and so for each m ∈ M ,

m = −(−m) ∈ M =⇒−m ∈ S,

and −m can only be in S if −− m ∈ M . So,

M = {m ∈ R such that − m ∈ S}.

Now we can apply the Mirror Proposition because M is the mirror image of S . By the Mirror Proposition
the greatest lower bound of M is −x .

♥

The set of missing least upper bounds is called the set of irrational numbers; it is

R \ Q.

How many of the real numbers are rational? We have seen that the rational numbers are countable.
However, we will prove that the real numbers are uncountable. This means that there are more irrational
numbers than there are rational numbers. These irrational numbers are a bit mysterious… In the vast
playground of the real numbers, we can think of the rational numbers as our trusted friends, who are
always there when we need them. The lemma below is called the Friendly Q Lemma, because it shows

that between any two real numbers, there is always a rational number. So, like a good friend, the rational
numbers Q are always close by when we need them.

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