Page 176 - 'Blast_Into_Math

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

and

|x| < 1.

There are two directions:

n
lim x =0 =⇒|x| < 1,
n→∞

and

n
|x| < 1=⇒ lim x =0.
n→∞

To prove the first direction, we assume that the geometric sequence converges to 0, and then we need to
prove that |x| < 1. We can prove this using the contrapositive. We assume that the conclusion, |x| < 1

is false. If this is false, then |x|≥ 1. So for each n ∈ N,

n
|x |≥ 1.

In the definition of limit with the ghost number =1/2 > 0, there is no N ∈ N such that for all
n> N ,

1
n
n
|x − 0| = |x | < = .
2
This means that the sequence does not converge to 0. We have proven (not B) implies (not A), where

A is the statement: the geometric sequence converges to 0, and B is the statement: |x| < 1. By the
Contrapositive Proposition, this proves the statement A implies B, which is: if the geometric sequence
converges to 0, then |x| < 1.

To prove the second direction, we start by assuming |x| < 1. A ghost number > 0 floats by. We need
to find a giant number N ∈ N to squash the first N sequence ants so that the surviving sequence ants
are trapped by the ghost between − and . Since x ∈ R , it’s possible that x/∈ Q . Since we need to
find N ∈ N , it makes sense to compare x to a rational number, because rational numbers are quotients

of integers, and the giant number N is also an integer. By the Friendly Q Lemma, there is always a
rational number nearby to help us.

By the Friendly Q Lemma, there exists a rational number z = p/q such that

p
|x| < < 1.
q

176