Page 156 - 'Blast_Into_Math

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

so

0 < .
|s| +1

Then,

˜ =
|s| +1

is also a ghost number, because

˜ > 0.

You can pronounce ˜ as either “epsilon-twiddle” or “epsilon-tilde.” For this ghost number there is a giant
˜
number N ∈ N (“N twiddle” or “N tilde”) such that
˜
|x n − X| < ˜ ∀n> N.

Then,

˜
|t n − sX| = |s||x n − X| < |s|˜ = |s| <, ∀n> N.
|s| +1

˜
So for the ghost number > 0, the giant number N ∈ N works with in the definition for
lim t n = sX.
n→∞

∞ also converges, so by the definition of the sequence
Now, we assume that the sequence {y n } n=1
converging, there exists a giant number M ∈ N such that for all n ∈ N with n> M,

|y n − Y | <.

This means that the distance between y n and Y is less than . Now, let’s think about z n = x n + y n .
We know that the distance between x n and X is less than for all n> N, and the distance between
y n and Y is less than for all n> M . We can write

|z n − (X + Y )| = |x n + y n − X − Y | = |x n − X + y n − Y |.

In geometry, you have learned the triangle inequality, which states that for any real numbers a , b , and c ,

TheTriangleInequality: |a − b|≤ |a − c| + |c − b|.

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