Sequences

Maggie Schildt

21 Sequences

In the previous chapter on calculus we began the discussion of sequences but we were not able to cover sequences convergence and divergence at the time. We will cover that here, as well recall that a sequence is an ordered set of terms, lets denote them $a_{n}$ , for all $a_{n}\in \mathbb{R}$ . Now in calculus we mentioned the idea of convergence of a sequence but we did not define it. Below we are given the formal definition as it is provided in analysis [18].

Definition.

A sequence $(x_{n})$ of real numbers converges to a limit $x\in \mathbb{R}$ , written

$\begin{equation*} x=\lim_{n\rightarrow \infty}{x_{n}},\ or\ x_{n}\rightarrow \infty \ as\ n\rightarrow \infty, \end{equation*}$

if for every $\epsilon > 0$ there exists $N\in \mathbb{N}$ such that

$\begin{equation*} |x_{n}-x|<\epsilon \ for\ all\ n>N. \end{equation*}$

It should also be noted that any sequence that does not converge, diverges, as the formal definition has been omitted.

If we want to analyze this definition, we already know that if a sequence is converging it is converging to a point $x$ . In this formal definition we now can see that this point can be calculated by taking the limit as $n$ goes to infinity of the sequence. This is possible only if the limit point is within a range such that the absolute valued difference between a term in the sequence and the limit point is less than some value, denoted by $\epsilon$ , greater than 0.

Now we need to provide a definition of $\epsilon$ in order to fully comprehend convergence.

Definition.

Epsilon, denoted $\epsilon$ is a positive, infinitesimal quantity whose limit is usually taken as $\epsilon\Rightarrow 0$ [39].

So when we tie this into our convergence definition for sequences the absolute valued difference of a term in the sequence and the limit point should be less than $\epsilon$ which should be decreasing to 0. This means that the difference should be getting smaller as such that $\epsilon \rightarrow 0$ .

Now that we have a good understanding of convergences for sequences lets look at monotonically increasing and monotonically decreasing sequences and their convergence as well as sub-sequences and convergence in the following subsections.

18.1 Monotone and Cauchy Sequences

In an analysis course we are asked to look past calculated results to patterns and be able to categorize functions based on those patterns. With sequences we can look at the pattern of the terms to describe if a sequence is monotone or not and if so, is the sequence monotonically increasing or monotonically decreasing.

A sequence is considered to be monotonically increasing if the the terms, defined as $x_{n}$ follow the relationship $x_{n}\leq x_{n+1}$ for all $n\in \mathbb{N}$ [2]. This means that the sequence is strictly increasing, every term is equal to or larger than the one previous.

We also have a monotonically decreasing sequence which is defined as a sequence with terms $x_{n}$ following the relationship of $x_{n}\geq x_{n+1}$ for all $n\in \mathbb{N}$ . This means that the sequence is strictly decreasing, every term is smaller than or equal to the term previous. Lets look at an example.

Example:

Consider the sequence $S_{n}=n^2$ for all $n\in \mathbb{N}$ . Let $n+1$ for $n+1\in \mathbb{N}$ be arbitrary, then $(n+1)^{2}\geq n^{2}$ . If we take the square root of both sides we get $\sqrt{(n+1)^{2}}\geq \sqrt{n^{2}}$ which simplifies to $n+1\geq n$ thus we can conclude that the sequence $S_{n}=n^{2}$ is monotonically increasing.

We briefly discussed the Axiom of Completeness in an earlier section which introduced the idea of bounded sets. Let us consider that sequences can be bounded, as well, either by a lower bound, an upper bound, or both.

Definition. Lower Bound

A sequence $x_{n}$ is said to contain a lower bound $a$ if $a\leq x_{n}$ [3].

Definition. Upper Bound

A sequence $x_{n}$ is said to contain an upper bound $b$ if $b\geq x_{n}$ [3].

We include these definitions because sequences can be bounded which means a sequence has both an upper and lower bound, $a\leq x_{n}\leq b$ . This bounded property leads us to our theorem for monotone convergence [3].

Theorem: The Monotone Convergence Theorem

If a sequence is monotone and bounded, then it converges.

This is quite straight forward and easy to apply for sequences that qualify. It should be noted that not every sequence is monotone, a perfect example of a sequence that is not monotone is an alternating sequence. Lets analyze an example.

Example:

Consider the sequence $S_{n}=(-1)^{n}\cdot n^{2}$ here we have a sequence whose every other term will be negative and the pattern of the sequence is a rotation of negatives and positives for every other term, this fails the definition for a monotonic sequences.

This does not mean that in analysis we just consider these sequences to be non-monotonic and toss them to the side. There is a large focus on non-monotone sequences and in real analysis courses we include this focus under the Cauchy sequence.

Definition. Cauchy Sequence

A sequence can be called Cauchy if, given $\epsilon>0$ , there exists $N$ such that if $m,n>N$ then $|a_{m}-a_{n}|<\epsilon$ .

To summarize, a Cauchy sequence is such that the terms of the sequence become arbitrarily close together, so much so that the difference is less than a value of $\epsilon$ that will continue to approach 0 as the space between terms grows smaller. Lets look at an example that is not monotone.

Example:

Consider the sequence

$\begin{equation*} a_{n}=\frac{(-1)^{n}}{n^{2}}\ for\ n\geq 1 \end{equation*}$

The first few values of the sequence are as follows:

$\begin{equation*} \left\lbrace -\frac{1}{1},\ \frac{1}{4},\ -\frac{1}{9},\ \frac{1}{16},\ -\frac{1}{25} \right\rbrace \end{equation*}$

The sequence is Cauchy because it is converging to 0, which can tell because the denominator is growing larger than the numerator. The sequence is not monotone because it is neither increasing nor decreasing exclusively, given that the $n+1$ term changes between negative and positive values.

Continuing on our discussion of Cauchy sequences there are a few properties to keep in mind. Any sequence that converges is a Cauchy sequence and is therefore a bounded sequence. There is also a theorem known as the Cauchy criterion which states the following.

Theorem: The Cauchy Criterion

A sequence converges if and only if it is a Cauchy Sequence

Although this theorem seems very exclusive for sequences it is one of the most inclusive sequences for determining convergence. Consider that monotone convergence relies on the fact that the sequence is monotone and bounded where as Cauchy sequences include convergent sequences that fail to be monotone as well as including monotone sequences.

We also discussed earlier the idea of closed sets and how closed sets contain all of their limit points. Since Cauchy sequences are any sequence that converges to a limit point, then we must introduce the following property [3].

Theorem: Properties of Limit Points

A point $x\in \mathbb{R}$ is a limit point of a set $A$ if anf only if there exists a sequence $a_{n}$ contained in A with $a_{n}\neq x$ for all $n\in \mathbb{N}$ , and $\lim_{n\rightarrow \infty}{a_{n}}=x$ .

We will discuss limits more later but including this property now is necessary to help understand the example we are about to look at for Cauchy sequences which brings together our understanding of closed sets, Cauchy sequences, and limit points.

Example:

Prove that a set $F\subseteq \mathbb{R}$ is closed if and only if every Cauchy sequence contained in $F$ has a limit that is also an element of $F$ [3].

Proof: Consider that every Cauchy sequence in $F\subseteq \mathbb{R}$ has a limit that is also an element of $F$ . If $F$ is closed then all the limit points of $F$ are contained within the $F$ . Let $x\in \mathbb{R}$ represents a limit that is contained in $F$ . If we have the Cauchy sequence $x_{n}$ in $F$ and it converges to $\lim_{n\rightarrow \infty}{x_{n}}=x$ then $x$ is an element of $F$ . Let us assume that $x\notin F$ then $x_{n}$ does not converge to $x$ for any $n\in N$ since $F$ is closed. By The Properties of Limit Points $x$ is a limit point of $F$ and is therefore an element of $F$ . This is a contradiction so we conclude that $x\in F$ .

Now consider that every Cauchy sequence in $F$ has a limit also in $F$ , then by The Proporties of Limit Points $\lim_{n\rightarrow \infty}{a_{n}}=x$ and thus the limits of the Cauchy sequences are in $F$ . If we let $x$ be an arbitrary limit of $F$ then there is a sequence $x_{n}\in F$ such that $x_{n}\neq x$ but which converges to $x$ then we say that this sequence is Cauchy and since $x_{n}$ and $x$ are in $F$ then we conclude that $F$ is closed.

This is a great problem to consider as it tests the properties of closed sets and Cauchy sequences as well as our understanding of limits and how sequences converge to limit points which may or may not fall within a set, depending on the set properties. We can also utilize our understanding of convergence and limit points to prove sets are open or closed depending on the type of sequence contained within them.

Mastering convergence is vital to success in an analysis course and with this in mind we now move on to the next subsection which will break down convergence and the idea of sub-sequences.

18.2 Subsequences and Convergence

We already know that limits determine if terms of a sequence converge to a real number, or diverge which means the limit of the sequence is either $\infty$ or $-\infty$ and we used this knowledge to study Cauchy sequences. Now, we will look at sub-sequences and how convergence can be derived from their associated properties.

We begin by looking at the formal definition for a sub-sequence.

Definition.

Let $a_{n}$ be a sequence of real numbers, and $n_{k}$ is a sequence of natural numbers where $n_{k}< n_{k+1}$ , then the sequence $a_{n_{k}}$ is a sub-sequence of $a_{n}$ [40].

Recall from our discussion on sets, that a we defined a subset as a set whose elements are also elements of another set. The same idea holds for sub-sequences, if we have a monotonically increasing sequence for which every term is a term in a larger sequence containing more terms then we can say that this sequence is a sub-sequence.

Let’s now consider some theorems that help us understand sub-sequences and their relation to a sequence [3].

Theorem:

Let $S_{n}$ be a convergent sequence with limit point $x$ , then every sub-sequence $n_{k}$ converges to limit point $x$ as well.

What this theorem tells us is that all sub-sequences under a convergent sequence, must also converge to the same limit as the limit of convergent sequence. This leads us into the next theorem for the relationship between sequences and sub-sequences [3].

Theorem: The Bolzano-Weierstrass Theorem

Every bounded sequence has a convergent sub-sequence.

This theorem is extremely useful because not every bounded sequence is convergent but by the Bolzano-Weierstrass does tell us some part of the sequence, a sub-sequence, does converge even if the sequence itself does not.

We could go further and address more theorems and more relationships between sub-sequences and sequences but we will end the discussion here and move on to the final section of real analysis that will grow our understanding of limits past what we know from calculus as well as what we have gained from limits and convergence.

21 Sequences

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