Page 169 - 'Blast_Into_Math

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

Now let’s assume that the sequence is increasing and is bounded above. Then it has a least upper bound
X ∈ R . By the definition of limit, we need to show that for any > 0, there exists N ∈ N such that
for all n> N,

|x n − X| <.

Let’s think about this on the number line. We need to show that if a ghost number > 0 floats by, we
can find a giant number N ∈ N such that all the ants in the sequence x n with n> N are trapped
between X − and X + . Since X is an upper bound, we already know that

x n ≤ X ∀n ∈ N.
So, since > 0,

X< X + ,
and

x n ≤ X< X + ∀n ∈ N.

But now, we need to squash all the ants to the left of X − , so that for all n> N ,

X − < x n .

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