18. Sequences

e1. Precise Definiton of Limits

We previously said that \(\lim\limits_{n\to\infty}a_n=L\) means that the terms \(a_n\) get arbitrarily close to a finite number \(L\) as \(n\) gets arbitrarily large. We now make this more precise by explaining the use of the word arbitrarily. The idea we want is:

If we specify how close to \(L\) the terms \(\{a_n\}\) should be (called an output tolerance) then for the part of the sequence beyond some point (called a tail of the sequence), the terms are all within that tolerance. The output tolerance is denoted by epsilon, \(\varepsilon\), and the tail of the sequence begins with some term \(a_n\) whose index is \(n=N\).

Here is the precise formulation of this idea:

We say the limit of the sequence \(a_n\) exists and is equal to a finite number \(L\) and we write \[ \lim_{n\to\infty}a_n=L \] to mean:

For all \(\varepsilon \gt 0\), there is a positive integer, \(N\), such that

if \(n \gt N\) then \(|a_n-L| \lt \varepsilon\).

If a sequence has a limit \(L\), we say it is convergent and that it converges to \(L\).
If a sequence does not have a limit, we say it is divergent or that it diverges.

The quantity \(|a_n-L|\) is the amount \(a_n\) differs from \(L\) and we require it to be less than the output tolerance \(\varepsilon\). The condition \(|a_n-L| \lt \varepsilon\) can also be written as \[ L-\varepsilon \lt a_n \lt L+\varepsilon \] which also says \(a_n\) is within \(\varepsilon\) of \(L\). The condition \(n \ge N\) says we require this for terms \(a_n\) beyond \(a_N\) in the sequence.

We also need a precise definition of an infinite limit. The idea here is that the terms \(a_n\) get larger than any specified number \(M\) and after some point the terms stay larger than \(M\).

We say the limit of the sequence \(a_n\) is positive infinity and we write \[ \lim_{n\to\infty}a_n=\infty \] to mean:

For all \(M \gt 0\), there is a positive integer, \(N\), such that

if \(n \gt N\) then \(a_n \gt M\).

We say the limit of the sequence \(a_n\) is negative infinity and we write \[ \lim_{n\to\infty}a_n=-\infty \] to mean:

For all \(M \lt 0\), there is a positive integer, \(N\), such that

if \(n \gt N\) then \(a_n \lt M\).

We emphasize that to say the limit is positive or negative infinity does not say that the limit exists! It merely says the way in which it does not exist, i.e. the way in which it diverges.

On the next few pages, we will use these definitions to prove the propositions and theorems which were left unproven on the previous pages.

18. Sequences

e1. Precise Definiton of Limits

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