Math for Non-Geeks/Limit: Convergence and divergence

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In this chapter we will introduce the central concept of a limit in the context of sequences. We will discuss what the term convergence is and what it means for a sequence to be convergent. All of the important concepts of Analysis like continuity, derivatives and integrals can only be defined if we have a notion of limits and convergence. Thus the concepts introduced in this chapter will serve as the backbone of our entire discussion about Analysis.

Intuition behind the idea of convergence

File:Konvergente Folge (Vorstellung, Beispiele, Definition).webm File:Folgenkonvergenz - Quatematik.webm Before one goes about to find a precise mathematical definition of a new concept, it is always a good idea to first develop an intuition. Let us therefore consider the harmonic sequence ${\left(\frac{1}{n}\right)}_{n\in \mathbb{N}}$. The first few elements of the sequence are:

You can already tell that elements become smaller and go towards zero as we increase the index $n$. We can make the following intuitive assertions:

• The sequence will get arbitrary close to $0$.
• The bigger the index $n$, the more the member $a_n=\frac{1}{n}$ gets close to $0$.
• The sequence $a_n=\frac{1}{n}$ tends to $0$.
• The sequence $a_n=\frac{1}{n}$ will reach $0$ at infinity .
• …

All this intuitive statements, already give us an idea of what the limit of a sequence is. To give a first intuitive definition we would say: The limit is the thing (or in this case the value) that the sequence approaches as we go further and further, i.e as the index $n$ goes to infinity. In our case $0$ is the limit of the harmonic sequence ${\left(\frac{1}{n}\right)}_{n\in \mathbb{N}}$. We say that the harmonic sequence is convergent. Of course not all sequences are convergent and thus have a finite limit: Think about the natural numbers $\mathbb{N}$ as a sequence. They grow bigger and bigger and never get closer to any particular value. We say that the sequence of the natural numbers is divergent.

Finding a definition for the limit

First steps

In mathematics we need an exact definition of what we mean by a "limit" to be able to talk and reason precisely. We can find such a definition if we start with an intuitive idea and gradually concretize it until we have a rigorous mathematical definition. The process of concretion will go on until we have a definition that employs only previously defined terms. Let's start with the following intuitive description of a limit:

But what does „arbitrary close" mean in the above sentence? We could translate as follows: Imagine the members of the sequence in a coordinate system, where the $x$-axis are the indices $n$ and the $y$-axis has the members/values of the sequence $a_n$. So every member of the sequence corresponds to a point in this coordinate system. The limit $a$ is marked by a dotted line.

If the members get "arbitrary close" to $a$ then the distance to the limit will keep getting smaller. Try to visualize a really narrow "hose" (maybe think about a garden hose) that has a radius of $ϵ$. Imagine taking this hose and "threading" it over the limit from the right. As long as the distance of the members from the limit is smaller than the thickness of the hose, we can keep pushing the hose to the left. All points are still inside the hose. But as soon as a point has a bigger distance from the limit, this point will not be inside the hose. At this point we stop threading the hose.

The point $a_{N_0}$ is the member of the sequence, from which on all members (so those with index greater than or equal $N_0$) will be inside the hose. Directly before $a_{N_0}$ there is a point that is outside of the hose (even by just a tiny amount). If we make the hose smaller, maybe some of the points that were inside the bigger one, now will be outside, and thus if your goal is to capture all points inside you can't "push" the hose as far as the bigger one. But even with this smaller hose it's still possible to capture almost all points:

The points that don't fit into the smaller hose are now more to the right than in the previous image. Let's call the first new member in the hose $a_{N_1}$. All members with an index greater or equal to $N_1$ will be inside the smaller hose.

Of course this "garden hose" has no mathematical meaning. We used it as a mental tool to show that the members of the sequence are closer to the dotted line the bigger the index $n$. They keep approaching $a$ and will indeed not start drifting away from the dotted line, because we have seen that all members starting from a certain index will be inside the hose, no matter how small we choose to make it. If we understood that, we don't need the "hose" anymore. We will replace our arbitrary small "hoses" with radius $ϵ$ with a mathematical object we call a $ϵ$-neighbourhood (it sounds more scary than it really is).

Every neighbourhood around the limit contains almost all members

We have found indices $N_0$ or $N_1$, from which on all later members of the sequence will be contained in the respective $ϵ$-hose. If we make the hose smaller then we can find another $N_2$, from which on all members will be inside the hose and so on. No matter how small we choose our hose, we will always find a point so that all subsequent points are captured inside that "hose".

Since all this starting indices $N_0$ are natural numbers, there are only a finite amount of members that are outside of the hose ($N_0-1$ to be precise). All other points are inside the hose. Since a sequence has infinitely many members, we can "ignore" the finite amount of members that are outside and say that "almost all" members are inside the hose. This is still true, even if we choose $N_0$ to be really big. Compared to infinity a finite amount – no matter how big you choose $N_0$ - is still little. Understanding this is important to understand the idea of limits.

Let's repeat this one more time: No matter how small you choose your hose to be, almost all members will be captured inside it. That means that the members will approach $a$. And this is the central idea about the limit. The members $a_n$ will get arbitrary close to $a$, if we choose a sufficiently large index $n$.

What is a neighbourhood of a number?

We can construct the neighbourhood of a number $a$ geometrically with a circle. Let $a$ be the center of a circle with radius $0$. Then we have only marked the point $a$ on the number line. If we increase the radius of the circle, we see that the diameter expands from $a$.

On an intuitive level a neighbourhood is set of numbers that encloses $a$.

In one dimension this circle is just an open interval. A neighbourhood of a number $a$ can be mathematically described by such an interval. The radius of the circle is the distance to the left and right boundary of the interval . The radius is a arbitrary (small) positive number $ϵ>0$.

An interval of this sort is characterized as the set of all numbers that are less than $ϵ$ apart from $a$. Thus this intervals will have the form $(a-ϵ,a+ϵ)$.

We call this interval $ϵ$-neighbourhood of $a$, and it looks like this:

The $ϵ$-neighbourhood of $a$ will more generally define a neighbourhood of $a$. A set $M$ is a neighbourhood of $a$, if and only if there exists an $ϵ$-neighbourhood $(a-ϵ,a+ϵ)$, such that $(a-ϵ,a+ϵ)\subseteq M$. Let's see how this works by considering the following set $M$:

First we need to find an $ϵ$-neighbourhood of $a$. Choose an adequate $ϵ>0$ and draw a circle with radius $ϵ$ around $a$. So that will mark an interval $(a-ϵ,a+ϵ)$. The set $M$ is also an interval. It encloses the interval $(a-ϵ,a+ϵ)$. Thus $M$ is a superset of the $ϵ$-neighbourhood of $a$. So our definition says that $M$ is a neighbourhood of $a$.

What does „almost all" mean?

To better understand this idea imagine a coordinate system in which we have infinitely many members of a convergent sequence with limit $a$. We thread a small $ϵ$-hose from the right over the limit. Then there are only a finite amount of members which are outside the hose, because their distance to $a$ is not small enough. But infinitely many members are inside the interval $(a-ϵ,a+ϵ)$ and thus in the $ϵ$-hose.

The amount of members that are inside the interval $(a-ϵ,a+ϵ)$ is overwhelmingly larger than the ones outside. Therefore is reasonable to say that almost all members are inside $(a-ϵ,a+ϵ)$.

Another way of expressing this idea would be to say that all but finitely many members are contained in the $ϵ$-hose.

A mathematical definition for the limit

Now that we have an idea about what we are trying to define, we will try to find a rigorous mathematical definition. We start by observing that:

We could already work with this definition. But for practical purposes it's useful to further formalise this definition.

Note that a member $a_n$ is an element of $(a-ϵ,a+ϵ)$, if and only if $|a_n-a|<ϵ$. Thus:

The part with „there exists a member, starting from which…“ can be equivalently formulated as „there exists a natural number $N$, so that for all $a_n$ with $n\ge N$ follows that …“. Thus:

This is the mathematical definition of the limit.

Definition of limit

This is what the individual parts of the above formula mean:

There are a few other important definitions when studying limits:

Convergence

A sequence is convergent, if the sequence has a limit. We say that the sequence converges towards $a$, if there is a limit $a$.

Divergence

If there is no limit a sequence is called divergent. Thus a sequence is divergent if it is not convergent.

Null Sequence

A sequence that has a limit of $0$ is called a Null Sequence.

If a sequence converges to $a$, we also write it like $\underset{n\to \mathrm{\infty }}{\lim }{a}_n=a$ or „$a_n\to a$ für $n\to \mathrm{\infty }$“. We say „Limes of $a_n$ for $n$ goes to infinity, is $a$“.

Explanation of convergence

Besides the above derivation of limits there is another intuition for the limit: The quantity $|a_n-a|$ is the distance between the n-th member and $a$. It is a measure of the error between $a_n$ and $a$. The inequality $|a_n-a|<ϵ$ means, that the error between $a_n$ and $a$ is guaranteed to be smaller than $ϵ$. Therefore we can interpret the definition of limit as follows: No matter how small we choose our maximum error $ϵ>0$ to be, almost all members are less than $ϵ$ away from the limit $a$. The error $|a_n-a|$ between the members and the limit becomes arbitrary small.

There are also historical reasons for this interpretation. Augustin-Louis Cauchy, who first proposed this definition, maybe wanted $ϵ$ to remind of the french word „erreur“ which means „error"..

Example: Convergence of the harmonic progression

Let's see this concepts in action by studying the harmonic progression with the generic member $a_n=\frac{1}{n}$. It should be intuitive that this sequences converges towards $0$. If this is correct, then this sequences should satisfy the definition for convergence towads $0$.

Take for example $ϵ=\frac{1}{2}$. Starting from the third member $a_3=\frac{1}{3}$ the distance between $a_n$ and $0$ is smaller than $ϵ=\frac{1}{2}$. All later members of this sequence are contained inside the $ϵ$-neighbourhood of $]-\frac{1}{2},\frac{1}{2}[$. For $ϵ=\frac{1}{10}$ this is true starting from $n=11$ and for $ϵ=0,00001$ starting from $n=100001$.

But the definition of limit requires that this is true for all $ϵ$. Therefore, let us assume we are given an $ϵ>0$. From the archimedean axiom it follows that there exists an $N\in \mathbb{N}$, so so that $\frac{1}{n}<ϵ$ for all $n\ge N$ (See Archimedean Axiom whit the choices of $x=1$ and $y=\frac{2}{ϵ}$.) From this $N$ onwards all members are contained inside the $ϵ$-neighbourhood $]-ϵ,ϵ[$. The definition tells us that $0$ is the limit of the harmonic progression.

The limit is unique

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