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

Let’s follow these steps to prove that

1
lim =0.
n→∞ n

1. In this example, the n term in the sequence is 1/n , so we can write the sequence as
th
1
∞
{x n } n=1 , x n = .
n

2. The role of L in the definition of the limit of the sequence is played by 0.
3. A ghost number > 0 floats by…
4. The ghost number needs to find a giant number N ∈ N to squash all the “ants” up to the

N th one, so that the ghost can then trap all the surviving ants between L − and L + .
Mathematically, the ghost needs to find a giant number N ∈ N such that

∀n> N, |x n − L| <.

5. Since L =0 in this example, the ghost number > 0 is looking for a giant number so that

∀n> N, |x n − 0| <.

Now, let’s use the definition of x n =1/n . The ghost number needs to find N ∈ N such that

1

∀n> N, − 0 <.

n

This is the same as
1
∀n> N, <.
n

With a bit of algebra, we see that this is the same as

1
∀n> N, <n.

6. So, the giant number N can be any natural number that is greater than 1/.
7. We have just proven that, for any ghost number > 0, we can find a giant number N ∈ N
such that

1

∀n> N, − 0 <.

n

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