Limits of Sequences, Limits of Sets

Karl Weierstrass (1815–1897).

In undergraduate mathematics, we are confronted at an early stage with “Epsilon-Delta” definitions. For example, given a function ${f(x)}$ of a real variable, we may ask what is the value of the function for a particular value ${x=a}$ . Maybe this is an easy question or maybe it is not.

The epsilon-delta concept can be subtle, and is sufficiently difficult that it has been used as a filter to weed out students who may not be considered smart enough to continue in maths (I know this from personal experience). The formulation of the epsilon-delta definitions is usually attributed to the German mathematician Karl Weierstrass. They must have caused him many sleepless nights.

In the simplest case, we just evaluate ${f(x)}$ at the point ${x=a}$ and get the answer ${f(a)}$ . But not all functions are so obliging. For example, the function

$\displaystyle f(x) = \frac{\sin x}{x}$

turns up early in the study of the calculus, when we calculate the derivative of the sine function. Substituting the value ${x=0}$ results in the indeterminate value ${f(0) = 0/0}$ . What to do now?

Limits of Functions

We look at the behaviour of ${f(x)}$ for values of ${x}$ arbitrarily close to ${x=0}$ . It is convenient to have a way of denoting the limiting value of ${f(x)}$ as ${x}$ approaches ${0}$ . This is usually written as ${\lim_{x\rightarrow 0} f(x)}$ . Now we define this limit to be the value ${f_0}$ such that the difference between ${f(x)}$ and ${f_0}$ becomes as small as we wish by choosing a value of ${x}$ that is sufficiently close to ${0}$ .

In symbolic terms, we introduce the epsilon-delta definition

$\displaystyle [ \lim_{x\rightarrow 0} f(x) = f_0 ] \iff [ \forall \epsilon > 0, \exists \delta > 0 : |x-0| < \delta \implies |f(x)-f_0| < \epsilon ] \,.$

John von Neumann(1903–1957).

At this point, I recall a response to the question “When do we really understand mathematics?”, addressed to the deep thinker John von Neumann. He is said to have answered “In mathematics, you never really understand anything, you just get used to it”. Certainly, it takes some time to get used to the epsilon-delta approach.

A very similar formulation us used to define the limit of a sequence ${\{x_n\}}$ . The convergence of the sequence to a value ${X}$ may be expressed in the form

$\displaystyle [ \lim_{x\rightarrow\infty} x_n = X] \iff [ \forall \epsilon > 0, \exists N\in\mathbb{N} : n>N \implies |x_n - X| < \epsilon ] \,.$

Here, the “small number” ${\delta}$ is replaced by the “large number” ${N}$ .

If all this is too much for you, check out the article “Topological Calculus: away with those nasty epsilons and deltas”, on this website (link below).

Limits of Sequences of Sets

So far, everything is standard, and should be familiar to any student of analysis. But the concept of limit is much more general. We look briefly at limits of sequences whose terms are not numbers, but sets.

Let ${\{A_n : n\in\mathbb{N} \}}$ be a sequence of sets. We will define two sets, the inferior and superior limits, or lower and upper limits:

$\displaystyle A_L = \liminf_{n\rightarrow\infty} A_n = \bigcup_{n\ge 1}^\infty \bigcap_{m\ge n}^\infty A_m \quad A_U = \limsup_{n\rightarrow\infty} A_n = \bigcap_{n\ge 1}^\infty \bigcup_{m\ge n}^\infty A_m$

These look complicated but can be described in simple terms:

x is in lim inf iff x is in all but a finite number of the sets A_n

x is in lim sup iff x is in an infinite number of the sets A_n

If the two sets ${A_L}$ and ${A_U}$ are equal, then the sequence has a limit, the common value of ${\liminf_{n\rightarrow\infty} A_n}$ and ${\liminf_{n\rightarrow\infty} A_n}$ :

$\displaystyle \lim_{n\rightarrow\infty} A_n = \bigcup_{n\ge 1}^\infty \bigcap_{m\ge n}^\infty A_m = \bigcap_{n\ge 1}^\infty \bigcup_{m\ge n}^\infty A_m \,.$

Probability Theory

Limits of this type play a crucial role in the theory of probability. In their wonderful book Ten Great Ideas About Chance, Persi Diaconis and Brian Skyrms posed the question: For an unlimited sequence of coin tosses, what are the chances of the probability of heads getting arbitrarily close to ${\frac{1}{2}}$ and remaining there? They wrote an expression for the corresponding set of points in the interval ${[0,1]}$ , which they introduced with a “drum roll”,

$\displaystyle \bigcap_{k=1}^\infty \bigcup_{n=1}^\infty \bigcap_{m\ge n}^\infty \left\{ x : \left| \frac{x_1+\cdots+x_m}{m}i - \frac{1}{2} \right| < \frac{1}{k} \right\} \,.$

The expression in braces is the set of all points in ${[0,1]}$ such that the average of the first ${m}$ coordinates is within ${1/k}$ of ${\frac{1}{2}}$ . The formula represents the strong law of large numbers.

Diaconis and Skyrms invited readers to examine this formula, which they described as “a horrible expression”. I believe they were joking: it is really beautiful, even if its attraction is what Bertrand Russell described as “a beauty cold and austere, like that of sculpture”.

Sources

${\bullet}$ Persi Diaconis and Brian Skyrms, 2018: Ten Great Ideas About Chance, Princeton Univ. Press. 255pp. ISBN: 9-780-6911-9639-8.

${\bullet}$ Wikipedia articles Set-theoretic limit and Limit inferior and limit superior. http://www.wikipedia.org/

${\bullet}$ ThatsMaths: Topological Calculus: away with those nasty epsilons and deltas. LINK.

ThatsMaths