Infinite Sequences

Historical Background

To better understand convergence — and ultimately, limits — it's best to put these ideas in context. The notion of convergence is all about control. What we want to be able to say, is: Given two real numbers ${n}$ and ${L,}$ after some point ${N,}$ the distance between ${n}$ and ${L}$ is so small, even if ${n \neq L.}$ Why do we need this control? Think about the sequence ${(1/n).}$ As ${n}$ gets extremely large, ${(1/n)}$ will get smaller and smaller. At some point, we get some number ${0.0 \ldots 01,}$ comprising so many zeros that we'd sooner die writing them before ever seeing the 1. We want to be able to say that ${(1/n)}$ converges to ${0.}$ That is, as ${x}$ gets very large (formally, "approaches infinity"), ${(1/n)}$ converges to 0.

It seems intuitive to say that ${(1/n)}$ converges to infinity, since we can see that ${(1/n)}$ gets smaller and smaller as ${n}$ gets larger and larger. But in mathematics, a thousand experiments do not make a theorem, nor do a million. The only way to firmly establish that ${(1/n)}$ converges to 0 is to prove it.

Now, why do we need sequences to converge? One reason is that without the notion of convergence, we'd never know with certainty if our approximations of functions are even remotely correct. We can never completely express ${\sqrt{2},}$ but we can approximate it with a function, assuming the function converges to ${\sqrt{2}.}$ But there are other reasons. Some mathematical statements are completely false if we don't have the notion of convergence. The sum ${\sum_{n=0}^{\infty} x^n}$ has the closed form fomula ${\frac{1}{1-x}.}$ Without the notion of convergence, we'd have ${1+2+4+8+\ldots=-1,}$ which is most definitely not what we want. Another reason is Zeno's Paradox: Suppose ${a}$ and ${b}$ are separated by 10 meters. To get from ${a}$ to ${b,}$ we must walk 5 meters. Then we must walk 2.5 meters. Then we must walk another 1.25, and so on. We never get to ${b!}$ We can solve these problems by defining the notion of convergence.¹.

When Newton and Leibniz began working on calculus, there was no notion of convergence. Instead, there were infinitesimals — numbers that lie between any two real numbers, but aren't real numbers themselves. For example, an infinitesimal lies between ${0.99999\ldots}$ and 1, but is not a real number. Almost immediately, we can see how this is difficult to think about. Mathematicians like Newton and Leibniz just "knew" when an infinitesimal would appear and disappear in their derivations. It was intuition that guided them, rather than some rigorous notion. The definition of convergence is aimed at making this idea more rigorous.

Formal Definition

If we think about the definition of a limit, the key requirement is "small." But the word "small" varies in meaning, depending on the listener and context. A thousand U.S. dollars might be small to a billionaire, but it certainly isn't small to those in abject poverty. Eight miles might seem small to marathon runners, but not to those who only run once every few years. Accordingly, we need a definition that appeases to everyone's idea of "small." Let's begin by defining some notation.

distance. Given ${x,y \in \reals,}$ we define the notation ${\d(y,x):=\abs{y-x}.}$

closeness. Given ${x,y,\varepsilon \in \reals}$ and ${\varepsilon \gt 0,}$ we say that ${x}$ and ${y}$ are close if and only if ${\d(x,y) \le \varepsilon.}$ If ${x}$ and ${y}$ are close, we write ${x \ec y.}$ Otherwise, we write ${x \not\ec y.}$

convergence. Let ${\varepsilon \gt 0}$ be a real number, let ${L}$ be a real number, and let ${\seq{a_n}{n=N}{\infty}}$ be a sequence of real numbers. We write ${\seq{a_n}{n=N}{\infty} \ec L}$ iff for all ${n \ge N,}$ it is true that ${a_n \ec L.}$ We say that a sequence ${\seq{a_n}{n=m}{\infty}}$ is eventually close to ${L}$ iff there exists an ${N \ge m}$ such that ${\seq{a_n}{n=N}{\infty} \ec L.}$ We say that ${\seq{a_n}{n=m}{\infty}}$ converges to ${L}$ iff ${\seq{a_n}{n=m}{\infty}}$ is eventually close to ${L.}$

It's helpful to view different implementations of this definition. Below is the standard definition found in most textbooks:

convergence. We say that a sequence ${(a_n)}$ converges to a number ${L \in \reals}$ if, and only if, for every ${\varepsilon \gt 0,}$ there exists an index ${N}$ such that, given any index ${n \ge N,}$ the relation ${\abs{a_n - L} \le \varepsilon}$ is true.

Let's be very clear about what each of these variables are. What is ${(a_n)?}$ ${(a_n)}$ is a sequence. It takes natural number indices, and pairs them up with real numbers — it establishes a specific ordering. What is ${L?}$ ${L}$ is a real number. It is not a term of the sequence (otherwise we wouldn't need to look for a limit in the first place). What is ${n?}$ ${n}$ is an index. What is ${N?}$ ${N}$ is also an index. What is ${\varepsilon?}$ ${\varepsilon}$ is a real number. What is ${a_n?}$ ${a_n}$ is a term of the sequence. What does it mean when ${(a_n)}$ converges? That there's a threshold index ${N}$ where all sequence's terms (starting at ${N}$) map are real numbers ${\varepsilon}$-close to ${L.}$ So, what does this tell us? If we want to prove that a sequence is divergent, we want to pick a ${\varepsilon}$ so small that ${a_n-L \gt \varepsilon.}$

example. The sequence ${S=\seq{(-1)^n}{n=0}{\infty}}$ is divergent. Assume ${S}$ is convergent. If ${S}$ is convergent, then ${S}$ converges to some ${L \in \reals.}$ That is, the terms of ${S}$ get arbitrarily close to ${L.}$ Fix ${\varepsilon=1.}$ Then we can choose an ${N \in \nat}$ such that:

Choose ${n = N.}$ Then ${\abs{(-1)^N - L \lt 1}}$ and ${\abs{(-1)^{N+1}-L}\lt 1,}$ since the former will yield 1, and the latter will yield -1 (the only two terms of ${S}$). It follows that ${\abs{1-L} \lt 1,}$ and ${-1-L \lt 1.}$ From the definition of the absolute value, we have

Transpose ${-1}$ to ${(a),}$ and we get ${0 \lt L.}$ Transpose ${1}$ to ${(b),}$ and we get ${L \lt 0.}$ We have a contradiction. ${L}$ cannot both be less than 0 and greater than 0. Our initial assumption is false, so ${S}$ must be divergent.