Page 153 - 'Blast_Into_Math

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

Since n> N and N> 1/ ,

1 1
n> so <.
n

Since

1
x n − 1= <,
n

1
|x n − 1| = <,
n

and for all “ants” x n with n> N ,

|x n − 1| <.

This fits precisely the definition of limit!

In general, you can follow these steps to prove that a sequence converges.

1. Write down the definition of the sequence

∞
{x n } n=1 ,

and the limit L .

2. A ghost number > 0 floats by…
3. You need to help the ghost number find a giant number N ∈ N such that for all n> N ,

|x n − L| <.

and
4. To do this, re-arrange the equation. Your goal is to arrive at an equation involving n> N,
which looks like
n> ∗∗∗

1
where *** is some expression involving , like .

5. The ghost number can choose any giant number N ∈ N such that N> ∗∗∗ .
6. Finally, you should always check your work and make sure that for all n> N ,

|x n − L| <.

153