Page 158 - 'Blast_Into_Math

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

Then if both n > N and n > M,

|x n − X| < and |y n − Y | <.

Exercise: What is a natural number that is bigger than both M and N ?

There are lots of giant numbers stomping around on the number line to the right of both M and N .
For example, N + M> N because M ∈ N means M ≥ 1, and for the same reason, N + M> M .
Let’s call the giant number G = N + M . We have shown that for all n> G ,

|x n + y n − (X + Y )|≤ |x n − X| + |y n − Y )| < + =2.

Since z n = x n + y n , we have proven that for any ghost number > 0 there exists a giant number
G ∈ N such that for all n> G ,

|z n − (X + Y )| < 2.

This is not quite the definition of limit, because there is a 2 in front of the . How can we fix this? Like

we did in the proof of (2), we can make a new ghost number. Since 2 =0,

˜ = > 0.
2

˜
˜
So, for the ghost number ˜ = /2, there also exists a giant number N ∈ N such that for all n> N ,
|x n − X| < ˜, |y n − Y | < ˜.

By the triangle inequality,

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

Now, with X + Y playing the role of L in the definition of limit, we have proven that the sequence

∞ converges to X + Y .
{z n } n=1

♥

7.1.2 Monotone sequences: marching mathematical ants

Monotone sequences are useful because they are orderly: the ants in a monotone sequence all march in
the same direction.

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