Chapter 5 Limits of sequences

5.1 Convergence and limte laws

Definition 5.1 (Distance between two real numbers) Given two real number $x$ and $y$, we define their distnce $d(x,y)$ to be $d(x,y) :=|x-y|$.

Definition 5.2 ($\varepsilon$-close real numbers) Let $\varepsilon$ be a real number. We say that two real numbers $x,y$ are $\varepsilon$-close iff we have $d(x,y)\leq \varepsilon$.

Definition 5.3 (Cauchy sequences of reals) Let $\varepsilon>0$ be a real number. A sequence $(a_n)_{n=N}^{\infty}$ of real numbers starting at some integer index $N$ is said to be $\varepsilon$-steady iff $a_j$ and $a_k$ are $\varepsilon$-close for every $j,k\geq N$. A sequence $(a_n)_{n=m}^{\infty}$ starting at some integer index $m$ is said to be eventually $\varepsilon$-steady iff there exists an $N\geq m$ such that $(a_n)_{n=N}^{\infty}$ is $\varepsilon$-steady. We say that $(a_n)_{n=m}^{\infty}$ is a Cauchy sequences iff it is eventually $\varepsilon$-steady for every $\varepsilon>0$.

Proposition 5.1 Let $(a_n)_{n=m}^{\infty}$ be a sequence of rational numbers starting at some integer index $m$. Then $(a_n)_{n=m}^{\infty}$ is a Cauchy sequence in the sense of Definition 5.4 iff it is a Cauchy sequence in the sense of Definition 6.3.

One can prove this with Proposition 5.7.

Definition 5.4 (Convergence of sequences) Let $\varepsilon>0$ be a real number, and let $L$ be a real number. A sequence $(a_n)_{n=N}^{\infty}$ of real numbers is said to be $\varepsilon$-close to $L$ iff $a_n$ is $\varepsilon$-close to $L$ for every $n\geq N$. A sequence $(a_n)_{n=m}^{\infty}$ is eventually $\varepsilon$-close to $L$ iff there exists an $N\geq m$ such that $(a_n)_{n=N}^{\infty}$ is $\varepsilon$-close to $L$. We say that $(a_n)_{n=m}^{\infty}$ converges to $L$ iff it is eventually $\varepsilon$-close to $L$ for every real $\varepsilon>0$.

Proposition 5.2 (Uniqueness of limits) Let $(a_n)_{n=m}^{\infty}$ be a real sequence starting at some integer index $m$, and let $L\ne L'$ be two distinct real numbers. Then it is not possible for $(a_n)_{n=m}^{\infty}$ to converge to $L$ while also converging to $L'$.

Definition 5.5 (Limits of sequences) If a sequence $(a_n)_{n=m}^{\infty}$ converges to some real number $L$, we say that $(a_n)_{n=m}^{\infty}$ is convergent and that its limit is $L$; we write \[ L= \lim_{n\to\infty}a_n \] to denote this fact. If a sequence $(a_n)_{n=m}^{\infty}$ is not converging to any real number $L$, we say that sequence $(a_n)_{n=m}^{\infty}$ is divergent and we leave $\lim_{n\to\infty}a_n$ undefined.

Proposition 5.3 (Convergent sequences are Cauchy) Suppose that $(a_n)_{n=m}^{\infty}$ is a convergent sequence of real numbers. Then $(a_n)_{n=m}^{\infty}$ is also a Cauchy sequence.

Now we show that formal limits can be superseded by actual limits, just as formal subtraction was superseded by actual subtraction when constructing the integers, and formal division superseded by actual division when constructing the rational numbers.

Proposition 5.4 (Formal limits are genuine limits) Suppose that $(a_n)_{n=1}^{\infty}$ is a Cauchy sequence of rational numbers. Then $(a_n)_{n=1}^{\infty}$ converges to ${\rm LIM}_{n\to \infty}a_n$, i.e. \[ {\rm LIM}_{n\to \infty}a_n = \lim_{n\to\infty}a_n \]

First we need to prove that the sequence $(a_n)_{n=m}^{\infty}$ is convergent, which is not an easy job. Then we can prove that it converges to ${\rm LIM}_{n\to \infty}a_n$.

Definition 5.6 (Bounded sequences) A sequence $(a_n)_{n=m}^{\infty}$ of real numbers is bounded by a real number $M$ iff $|a_n|\leq M$ for all $n\geq m$. We say that $(a_n)_{n=m}^{\infty}$ is bounded iff it is bounded by $M$ for some rational $M>0$.

Corollary 5.1 Every convergent sequence of real numbers is bounded.

Theorem 5.1 (Limit Laws) Let $(a_n)_{n=m}^{\infty}$ and $(b_n)_{n=m}^{\infty}$ be convergent sequences of real numers, and let $x,y$ be the real numbers $x:=\lim_{n\to\infty} a_n$ and $y:\lim_{n\to\infty} b_n$.

\[\lim_{n\to\infty}(a_n+b_n)=\lim_{n\to\infty}a_n+\lim_{n\to\infty}b_n.\]
\[\lim_{n\to\infty}(a_nb_n)=(\lim_{n\to\infty}a_n)(\lim_{n\to\infty}b_n).\]
\[\lim_{n\to\infty}(ca_n)=c\lim_{n\to\infty}a_n.\]
\[\lim_{n\to\infty}(a_n-b_n)=\lim_{n\to\infty}a_n-\lim_{n\to\infty}b_n.\]
Suppose that $y\ne 0$, and that $b_n\ne 0$ for all $n\geq m$. Then \[ \lim_{n\to\infty}b_n^{-1} = (\lim_{n\to\infty}b_n)^{-1}. \]
Suppose that $y\ne 0$, and that $b_n\ne 0$ for all $n\geq m$. Then \[ \lim_{n\to\infty}\frac{a_n}{b_n} = \frac{\lim_{n\to\infty}a_n}{\lim_{n\to\infty}b_n}. \]
\[\lim_{n\to\infty}\max(a_n,b_n)=\max(\lim_{n\to\infty}a_n,\lim_{n\to\infty}b_n).\]
\[\lim_{n\to\infty}\min(a_n,b_n)=\min(\lim_{n\to\infty}a_n,\lim_{n\to\infty}b_n).\]

5.2 The Extended real number system

Definition 5.7 (Extended real number system) The extended real number system $\mathbb{R}^{\ast}$ is the real line $\mathbb{R}$ with two additional elements attached, called $+\infty$ and $-\infty$. These elements are distinct from each other and also distinct from every real number. An extended real number $x$ is called finite iff it is a real number, and infinite iff it is equal to $+\infty$ or $-\infty$.

Definition 5.8 (Negation of extended reals) The operation of negation $x\mapsto -x$ on $\mathbb{R}$, we now extend to $\mathbb{R}^{\ast}$ by defining $-(+\infty):=-\infty$ and $-(-\infty):=+\infty$.

Definition 5.9 (Ordering of extended reals) Let $x$ and $y$ be extended real numbers. We say that $x\leq y$, i.e., $x$ is less than or equal to $y$, iff one of the following three statements is true:

$x$ and $y$ are real numbers, and $x\leq y$ as real numbers.
$y=+\infty$.
$x=-\infty$.

We say that $x<y$ if we have $x\leq y$ and $x\ne y$. We sometimes write $x<y$ as $y>x$, and $x\leq y$ as $y\geq x$.

Proposition 5.5 Let $x,y,z$ be extended real numbers. Then the following statements are true:

(Reflexivity) We have $x\leq x$.
(Trichotomy) Exactly one of the statements $x<y$ ,$x=y$, or $x>y$ is true.
(Transitivity) If $x\leq y$ and $y\leq z$, then $x\leq z$.
(Negation reverses other) If $x\leq y$, then $-y\leq -x$.

Definition 5.10 (Supremum of sets of extended reals) Let $E$ be a subset of $\mathbb{R}^{\ast}$. Then we define the supremum $\sup(E)$ or least upper bound of $E$ by the following rule.

If $E$ is contained in $\mathbb{R}$, then we let $\sup(E)$ be as defined in Definition 5.19
If $E$ contains $+\infty$, then we set $\sup(E):=+\infty$.
If $E$ does not contain $+\infty$ but does contain $-\infty$, then we set $\sup(E):=\sup(E\setminus\{-\infty\})$

We also define the infimum $\inf(E)$ of $E$ (also known as the greatest lower bound of $E$) by the formula \[ \inf(E) :=-\sup(-E) \] where $-E$ is the set $-E:=\{-x:x\in E\}$.

Let $E$ be the empty set. Then $\sup(E) = -\infty$ and $\inf(E) = +\infty$. This is the only case in which the supremum can be less than the infimum.

Theorem 5.2 Let $E$ be a subset of $\mathbb{R}^{\ast}$. Then the following statements are true.

Forevery $x\in E$ we have $x\leq \sup(E)$ and $x\geq \inf(E)$.
Suppose that $M\in \mathbb{R}^{\ast}$ is an upper bound for $E$, i.e., $x\leq M$ for all $x\in E$. Then we have $\sup(E)\leq M$.
Suppose that $M\in \mathbb{R}^{\ast}$ is an lower bound for $E$, i.e., $x\leq M$ for all $x\in E$. Then we have $\inf(E)\geq M$.

5.3 Suprema and Infima of sequences

Definition 5.11 (Sup and inf of sequences) Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers. Then we define $\sup(a_n)_{n=m}^{\infty}$ to be the supremum of the set $\{a_n:n\geq m\}$, and $\inf(a_n)_{n=m}^{\infty}$ to be the infimum of the same set $\{a_n:n\geq m\}$. The quantities $\sup(a_n)_{n=m}^{\infty}$ and $\inf(a_n)_{n=m}^{\infty}$ are sometimes written as $\sup_{n\geq m}a_n$ and $\inf_{n\geq m}a_n$ respectively.

Proposition 5.6 (Least upper bound property) Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers, and let $x$ be the extended real number $x:=\sup(a_n)_{n=m}^{\infty}$. Then we have $a_n\leq x$ for all $n\geq m$. Also, whenever $M \in \mathbb{R}^{\ast}$ is an upper bound for $a_n$, we have $x\leq M$. Finally, for every extended real number $y$ for which $y<x$, there exists at least one $n\geq m$ for which $y< a_n\leq x$.

Proposition 5.7 (Monotone bounded sequences converge) Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers which has some finite upper bound $M \in \mathbb{R}$, and which is also increasing. Then $(a_n)_{n=m}^{\infty}$ is convergent, and in fact \[ \lim_{n\to\infty}a_n=\sup (a_n)_{n=m}^{\infty} \leq M \]

One can similarly prove that if sequence $(a_n)_{n=m}^{\infty}$ is bounded below and decreasing, then it is convergent, and that the limit is equal to the infimum.

Corollary 5.2 Let $0<x<1$. Then we have $\lim_{n\to \infty}x^n=0$. (Hint: let $L =\lim_{n\to \infty}x^n$, then we see that $(x^{n+1})_{n=1}^{\infty}$ converges to $xL$ from Theorem 5.1. But the sequence $(x^{n+1})_{n=1}^{\infty}$ is just the sequence $(x^{n})_{n=2}^{\infty}$. So $L=xL$.)

5.4 Limsup, Liminf, and limit points

Definition 5.12 (Limit points) Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers, let $x$ be a real number, and let $\varepsilon>0$ be a real number. We say that $x$ is $\varepsilon$-adherent to $(a_n)_{n=m}^{\infty}$ iff there exists an $n\geq m$ such that $a_n$ is $\varepsilon$-close to $x$. We say that $x$ is continually $\varepsilon$-adherent to $(a_n)_{n=m}^{\infty}$ iff it is $\varepsilon$-adherent to $(a_n)_{n=N}^{\infty}$ for every $N\geq m$. We say that $x$ is a limit point or adherent point of $(a_n)_{n=m}^{\infty}$ iff it is continually $\varepsilon$-adherent to $(a_n)_{n=m}^{\infty}$ for every $\varepsilon>0$.

Proposition 5.8 (Limits are limit points) Let $(a_n)_{n=m}^{\infty}$ be a sequence which converges to a real number $c$. Then $c$ is a limit point of $(a_n)_{n=m}^{\infty}$, and in fact it is the only limit point of $(a_n)_{n=m}^{\infty}$.

Definition 5.13 (Limit superior and limit inferior) Suppose that $(a_n)_{n=m}^{\infty}$ is a sequence. We define a new sequence $(a_N^+)_{N=m}^{\infty}$ by the formula \[ a_N^+:=\sup(a_n)_{n=N}^{\infty}. \] We define the limit superior of the sequence $(a_n)_{n=m}^{\infty}$, denoted $\limsup_{n\to\infty}a_n$, by the formula \[ \limsup_{n\to\infty}a_n := \inf (a_N^+)_{N=m}^{\infty}. \] Similarly, we can define \[ a_N^-:=\inf(a_n)_{n=N}^{\infty} \] and define the limit inferior of the sequence $(a_n)_{n=m}^{\infty}$, denoted $\liminf_{n\to\infty}a_n$, by the formula \[ \liminf_{n\to\infty}a_n := \sup (a_N^-)_{N=m}^{\infty}. \]

Note that the starting index $m$ of the sequence is irrelevant.

Proposition 5.9 Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers, let $L^+$ be the limit superior of this sequence, and let $L^-$ be the limit inferior of this sequence.

For every $x>L^+$, there exists an $N\geq m$ such that $a_n<x$ for all $n\geq N$. Similarly, for every $y<L^-$, there exists an $N\geq m$ such that $a_n>y$ for all $n\geq N$.
For every $x<L^+$, and every $N\geq m$, there exists an $n\geq N$ such that $a_n>x$. Similarly, for every $y>L^-$, and every $N\geq m$, there exists an $n\geq N$ such that $a_n<y$.
We have $\inf(a_n)_{n=m}^{\infty}\leq L^-\leq L^+\leq \sup(a_n)_{n=m}^{\infty}$
If $c$ is any limit point of $(a_n)_{n=m}^{\infty}$, then we have $L^-\leq c\leq L^+$.
If $L^+$ is finite, then it is a limit point of $(a_n)_{n=m}^{\infty}$. Similarly, if $L^-$ is finite, then it is a limit point of $(a_n)_{n=m}^{\infty}$.
Let $c$ be a real number. If $(a_n)_{n=m}^{\infty}$ converges to $c$, then we must have $L^+=L^-=c$. Conversely, if $L^+=L^-=c$, then $(a_n)_{n=m}^{\infty}$ converges to $c$.

Lemma 5.1 (Comparison principle) Suppose that $(a_n)_{n=m}^{\infty}$ and $(b_n)_{n=m}^{\infty}$ are two sequence of real numbers such that $a_n\leq b_n$ for all $n\geq m$. Then we have the inequalities \[ \begin{aligned} \sup (a_n)_{n=m}^{\infty} &\leq \sup (b_n)_{n=m}^{\infty}\\ \inf (a_n)_{n=m}^{\infty} &\leq \inf (b_n)_{n=m}^{\infty}\\ \limsup_{n\to\infty}a_n &\leq \limsup_{n\to\infty}b_n\\ \liminf_{n\to\infty}a_n &\leq \liminf_{n\to\infty}b_n \end{aligned} \]

Corollary 5.3 (Squeeze test) Let $(a_n)_{n=m}^{\infty}$, $(b_n)_{n=m}^{\infty}$, $(c_n)_{n=m}^{\infty}$ be sequences of real numbers such that $a_n\leq b_n\leq c_n$ for all $n\geq m$. Suppose also that $(a_n)_{n=m}^{\infty}$ and $(c_n)_{n=m}^{\infty}$ both converge to the same limit $L$. Then $(b_n)_{n=m}^{\infty}$ is also convergent to $L$.

Corollary 5.4 (Zero test for sequences) Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers. Then the limit $\lim_{n\to\infty}a_n$ exists and is equal to zero iff the limit $\lim_{n\to \infty}|a_n|$ exists and is equal to zero.

Theorem 5.3 (Completeness of the reals) A sequence $(a_n)_{n=1}^{\infty}$ of real numbers is a Cauchy sequence iff it is convergent.

Note that while it is very similar in spirit to Proposition 5.4, it is a bit more general. Here we can prove this easily since we only need to show that $L^-=L^+$ according to 5.9.

Theorem 5.3 asserts that the real numbers are a complete metric space - that they do not contain “holes” the same way the rationals do. (Certainly the rationals have lots of Cauchy sequences which do not converge to other rationals) This property is closely related to the least upper bound property (Theorem 5.1), and is one of the principal characteristics which make the real numbers superior to the rational numbers for the purposes of doing analysis (taking limits, taking derivatives and integrals, finding zeroes of functions, that kind of thing), as we shall see in later chapters.

5.5 Subsequences

Definition 5.14 Let $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ be sequences of real numbers. We say that $(b_n)_{n=0}^{\infty}$ is a subsequence of $(a_n)_{n=0}^{\infty}$ iff there exists a function $f:\mathbb{N}\to\mathbb{N}$ which is strictly increasing such that \[ b_n=f(a_n) \text{ for all } n\in\mathbb{N}. \]

Lemma 5.2 Let $(a_n)_{n=0}^{\infty}$, $(b_n)_{n=0}^{\infty}$, and $(c_n)_{n=0}^{\infty}$ be sequences of real numbers. Then $(a_n)_{n=0}^{\infty}$ is a subsequence of $(a_n)_{n=0}^{\infty}$. Furthermore, if $(b_n)_{n=0}^{\infty}$ is a subsequence of $(a_n)_{n=0}^{\infty}$, and $(c_n)_{n=0}^{\infty}$ is a subsequence of $(b_n)_{n=0}^{\infty}$, then $(c_n)_{n=0}^{\infty}$ is a subsequence of $(a_n)_{n=0}^{\infty}$.

Proposition 5.10 (Subsequences related to limits) Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers, and let $L$ be a real number. Then the following two statements are logically equivalent:

The sequence $(a_n)_{n=0}^{\infty}$ converges to $L$.
Every subsequence of $(a_n)_{n=0}^{\infty}$ converges to $L$.

Proposition 5.11 (Subsequences related to limit points) Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers, and let $L$ be a real number. Then the following two statements are logically equivalent.

$L$ is a limit point of $(a_n)_{n=0}^{\infty}$.
There exists a subsequence of $(a_n)_{n=0}^{\infty}$ which converges to $L$.

Theorem 5.4 (Bolzano-Weierstrass theorem) Let $(a_n)_{n=0}^{\infty}$ be a bounded sequence. Then there is at least one subsequence of $(a_n)_{n=0}^{\infty}$ which converges.

One can prove this with Proposition 5.9 and Proposition 5.11.

The Bolzano-Weierstrass theorem says that if a sequence is bounded, then eventually it has no choice but to converge in some places; it has “no room” to spread out and stop itself from acquiring limit points. In the language of topology, this means that the interval $\{x\in\mathbb{R}:-M\leq x\leq M\}$ is compact, whereas an unbounded set such as the real line $\mathbb{R}$ is not compact.

5.6 Real exponentiation, part II

Lemma 5.3 (Continuity of exponentiation) Let $x>0$, and let $\alpha$ be a real number. Let $(q_n)_{n=1}^{\infty}$ be any sequence of rational numbers converging to $\alpha$. Then $(x^{q_n})_{n=1}^{\infty}$ is also a convergent sequence. Furthermore, if $(q'_n)_{n=1}^{\infty}$ is any other sequence of rational numbers converging to $\alpha$, then $(x^{q'_n})_{n=1}^{\infty}$ has the same limit as $(x^{q_n})_{n=1}^{\infty}$: \[ \lim_{n\to \infty} x^{q_n}=\lim_{n\to \infty} x^{q'_n}. \]

Definition 5.15 (Exponentiation to a real exponent) Let $x>0$ be real, and let $\alpha$ be real number. We define the quantity $x^{\alpha}$ by the formula $x^\alpha = \lim_{n\to \infty} x^{q_n}$, where $(q_n)_{n=1}^{\infty}$ is any sequence of rational numbers converging to $\alpha$.

Proposition 5.12 All the results of Lemma 5.12, which held for rational numbers $q$ and $r$, continue to hold for real numbers $q$ and $r$.