2.1 Sequences and limits

A sequence of real or complex numbers is defined to be a function from the set $N$ of natural numbers into the set $R$ or $C.$ Instead of referring to such a function as an assignment $n\to f\left(n\right),$ we ordinarily use the notation $\left\{a_n\right\},$ ${\left\{a_n\right\}}_1^{\infty },$ or $\{a_1,a_2,a_3,...\}.$ Here, of course, $a_n$ denotes the number $f\left(n\right).$

REMARK We expand this definition slightly on occasion to make some of our notation more indicative.That is, we sometimes index the terms of a sequence beginning with an integer other than 1. For example, we write ${\left\{a_n\right\}}_0^{\infty },$ $\{a_0,a_1,...\},$ or even ${\left\{a_n\right\}}_{-3}^{\infty }.$

We give next what is the most significant definition in the whole of mathematical analysis, i.e., what it means for a sequence to converge or to have a limit.

Let $\left\{a_n\right\}$ be a sequence of real numbers and let $L$ be a real number. The sequence $\left\{a_n\right\}$ is said to converge to $L,$ or that $L$ is the limit of $\left\{a_n\right\}$ , if the following condition is satisfied.For every positive number $ϵ,$ there exists a natural number $N$ such that if $n\ge N,$ then $|a_n-L|<ϵ.$

In symbols, we say $L=lim{a}_n$ or

$$L=\underset{n\to \infty }{lim}{a}_n.$$

We also may write $a_n\mapsto L.$

If a sequence $\left\{a_n\right\}$ of real or complex numbers converges to a number $L,$ we say that the sequence $\left\{a_n\right\}$ is convergent .

We say that a sequence $\left\{a_n\right\}$ of real numbers diverges to $+\infty$ if for every positive number $M,$ there exists a natural number $N$ such that if $n\ge N,$ then $a_n\ge M.$ Note that we do not say that such a sequence is convergent.

Similarly, we say that a sequence $\left\{a_n\right\}$ of real numbers diverges to $-\infty$ if for every real number $M,$ there existsa natural number $N$ such that if $n\ge N,$ then $a_n\le M.$

The definition of convergence for a sequence $\left\{z_n\right\}$ of complex numbers is exactly the same as for a sequence of real numbers.Thus, let $\left\{z_n\right\}$ be a sequence of complex numbers and let $L$ be a complex number. The sequence $\left\{z_n\right\}$ is said to converge to $L,$ or that $L$ is the limit of $\left\{z_n\right\},$ if the following condition is satisfied.For every positive number $ϵ,$ there exists a natural number $N$ such that if $n\ge N,$ then $|z_n-L|<ϵ.$

REMARKS The natural number $N$ of the preceding definition surely depends on the positive number $ϵ.$ If ${ϵ}^{\text{'}}$ is a smaller positive number than $ϵ,$ then the corresponding $N^{\text{'}}$ very likely will need to be larger than $N.$ Sometimes we will indicate this dependence by writing $N\left(ϵ\right)$ instead of simply $N.$ It is always wise to remember that $N$ depends on $ϵ.$ On the other hand, the $N$ or $N\left(ϵ\right)$ in this definition is not unique. It should be clear that if a natural number $N$ satisfies this definition, then any larger natural number $M$ will also satisfy the definition. So, in fact, if there exists one natural number that works, then there exist infinitely many such natural numbers.

It is clear, too, from the definition that whether or not a sequence is convergent only depends on the “tail” of the sequence.Specifically, for any positive integer $K,$ the numbers $a_1,a_2,...,a_K$ can take on any value whatsoever without affecting the convergence of the entire sequence.We are only concerned with $a_n$ 's for $n\ge N,$ and as soon as $N$ is chosen to be greater than $K,$ the first part of the sequence is irrelevant.

The definition of convergence is given as a fairly complicated sentence, and there are several other ways of saying the same thing. Here are two:For every $ϵ>0,$ there exists a $N$ such that, whenever $n\ge N,$ $|a_n-L|<ϵ.$ And, given an $ϵ>0,$ there exists a $N$ such that $|a_n-L|<ϵ$ for all $n$ for which $n\ge N.$ It's a good idea to think about these two sentences and convince yourself that they really do “mean” the same thing as the one defining convergence.