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

In this case s plays the role of z in the lemma.

If y ∈ Q and x/∈ Q , then −x/∈ Q . This is because Q contains additive inverses, so if −x ∈ Q
then x = −− x ∈ Q , but x/∈ Q . So −x ∈ R is the least upper bound of a non-empty set T ⊂ Q .
By the mirror proposition, the set

M = {m ∈ R such that − m ∈ T}

is bounded below, and the greatest lower bound of M is −− x = x . Since m ∈ M means that
−m ∈ T ⊂ Q , −(−m)= m ∈ Q . So, M ⊂ Q . Since x is the greatest lower bound of M and
x< y , this means that y is not a lower bound for M . So there is some m ∈ M with

m< y.

Because x is a lower bound for M ,

x ≤ m.

Can x = m ? Well, x ∈ R \ Q but m ∈ M ⊂ Q , so x = m . Since x ≤ m this means that

x< m.

So,

x<m< y,

and in this case the role of z in the lemma is played by m.

♥

With the help of the real numbers and the friendly rational numbers who are always close by, we can
understand what happens to marching monotone mathematical ants.

Proposition 7.2.5 (Marching Ant Proposition). Let

∞
{x n } n=1

be a monotone sequence. If it is increasing, then the sequence converges if and only if the sequence is bounded
above, in which case, the limit of the sequence is its least upper bound. If the sequence is decreasing, then it
converges if and only if it is bounded below, in which case the limit of the sequence is its greatest lower bound.

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