A Course in Calculus and Real Analysis

Sudhir R. Ghorpade, Balmohan V. Limaye

Chapter 2

Sequences - all with Video Answers

Educators

Chapter Questions

Problem 1

Which of the following sequences are bounded? Which of them are convergent? In case of convergence, find the limit.
(i) $a_{n}:=\frac{1}{n^{2}}$,
(ii) $a_{n}:=\sqrt{n}$,
(iii) $a_{n}:=(-1)^{n}$,
(iv) $a_{n}:=\frac{n}{2 n+1}$,
(v) $a_{n}:=\sqrt{n}(\sqrt{n+1}-\sqrt{n})$,
(vi) $a_{n}:=n^{3 / 2}\left(\sqrt{n^{3}+1}-\sqrt{n^{3}}\right)$.

Wendi Zhao

Wendi Zhao

Numerade Educator

Problem 2

Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences in $\mathbb{R}$. Under which of the following conditions is the sequence $\left(a_{n} b_{n}\right)$ convergent? Justify.
(i) $\left(a_{n}\right)$ is convergent.
(ii) $\left(a_{n}\right)$ is convergent and $\left(b_{n}\right)$ is bounded.
(iii) $\left(a_{n}\right)$ converges to 0 and $\left(b_{n}\right)$ is bounded.
(iv) $\left(a_{n}\right)$ and $\left(b_{n}\right)$ are convergent.

Lucas Finney

Numerade Educator

Problem 3

Let $a, b \in \mathbb{R}$ and $\left(a_{n}\right)$ be a sequence in $\mathbb{R}$ such that $a_{n} \rightarrow a$. and $a_{n} \geq b$ for all $n \in \mathbb{N}$. Show that $a \geq b$. Give an example in which $a_{n}>a$ for all $n \in \mathbb{N}$, but $a_{n} \rightarrow a$.

Julian Wong

Numerade Educator

Problem 4

Let $a$ and $x$ be real numbers. If $\left(b_{n}\right)$ and $\left(c_{n}\right)$ are sequences in $\mathbb{R}$ such that
$$
\lim _{n \rightarrow \infty} b_{n}=0=\lim _{n \rightarrow \infty} c_{n} \quad \text { and } \quad a-b_{n} \leq x \leq a+c_{n} \quad \text { for } n \in \mathbb{N}
$$
then show that $x=a$.

Uma Kumari

Numerade Educator

Problem 5

If $\left(a_{n}\right)$ is a sequence in $\mathbb{R}$ such that $a_{n} \neq 0$ for all $n, \lim _{n \rightarrow \infty}\left|a_{n+1} / a_{n}\right|$ exists and it is less than 1, then show that $a_{n} \rightarrow 0$.

Mengchun Cai

Numerade Educator

Problem 6

If $k \in \mathbb{N}$ and $x \in \mathbb{R}$ with $|x|<1$, then show that
$$
\lim _{n \rightarrow \infty} n^{k} x^{n}=0.
$$

Khushbu Rani

Numerade Educator

Problem 7

For $n \in \mathbb{N}$, let $a_{n}:=n^{1 / n}$. Show that $a_{1}<a_{2}<a_{3}$ and $a_{n}>a_{n+1}$ for all $n \geq 3$. Further, show that
$$
1<a_{n}<1+\frac{\sqrt{2}}{\sqrt{n-1}} \text { for all } n \geq 2
$$
and deduce that $a_{n} \rightarrow 1$ as $n \rightarrow \infty$.

JH

J Hardin

Numerade Educator

Problem 8

Show that the sequence $\left(B_{n}\right)$ defined by
$$
B_{n}:=\left(1+\frac{1}{n}\right)^{n} \quad \text { for } n \in \mathbb{N}
$$
is monotonically increasing. Deduce that the sequence $\left(B_{n}\right)$ is convergent. (Hint: Given $n \in \mathbb{N}$, use the A.M.-G.M. inequality for $a_{1}=\cdots=a_{n}:=$ $1 /(n+1)$ and $a_{n+1}:=1$. Also, note that $B_{n} \leq 3$ for all $n \in \mathbb{N}$.)

JH

J Hardin

Numerade Educator

Problem 9

Show that the sequence $\left(a_{n}\right)$ is convergent and find its limit if $\left(a_{n}\right)$ is given by the following.
(i) $a_{1}:=1$ and $a_{n+1}:=\left(3 a_{n}+2\right) / 6$ for $n \in \mathbb{N}$.
(ii) $a_{1}:=1$ and $a_{n+1}:=a_{n} /\left(2 a_{n}+1\right)$ for $n \in \mathbb{N}$.
(iii) $a_{1}:=1$ and $a_{n+1}:=2 a_{n} /\left(4 a_{n}+1\right)$ for $n \in \mathbb{N}$.
(iv) $a_{1}:=2$ and $a_{n+1}:=\sqrt{1+a_{n}}$ for $n \in \mathbb{N}$.
(v) $a_{1}:=1$ and $a_{n+1}:=\sqrt{2+a_{n}}$ for $n \in \mathbb{N}$.
(vi) $a_{1}:=2$ and $a_{n+1}:=(1 / 2)+\sqrt{a_{n}}$ for $n \in \mathbb{N}$.
(vii) $a_{1}:=1$ and $a_{n+1}:=(1 / 2)+\sqrt{a_{n}}$ for $n \in \mathbb{N}$.

JH

J Hardin

Numerade Educator

Problem 10

For $n \in \mathbb{N}$, let
$$
a_{n}:=\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2 n} \quad \text { and } \quad b_{n}:=\frac{1}{1}+\frac{1}{3}+\cdots+\frac{1}{2 n-1}
$$
Show that $a_{n} \rightarrow \infty$ and $b_{n} \rightarrow \infty$. (Hint: Example 2.10 (iii).)

Lucas Finney

Numerade Educator

Problem 11

Show that $(n !)^{1 / n} \rightarrow \infty$

Vishnu P

Numerade Educator

Problem 12

Suppose $\alpha$ and $\beta$ are real numbers such that $0 \leq \beta \leq \alpha$. Let
$$
a_{1}:=\alpha, \quad b_{1}:=\beta \quad \text { and } \quad a_{n+1}:=\frac{a_{n}+b_{n}}{2}, b_{n+1}:=\sqrt{a_{n} b_{n}} \quad \text { for } n \in \mathbb{N}
$$
Show that $\left(a_{n}\right)$ is a monotonically decreasing sequence that is bounded below by $\beta$, and $\left(b_{n}\right)$ is a monotonically increasing sequence that is bounded above by $\alpha$. Further, show that $0 \leq \alpha-\beta \leq(\alpha-\beta) / 2^{n-1}$ for $n \in \mathbb{N}$. Deduce that $\left(a_{n}\right)$ and $\left(b_{n}\right)$ are convergent and have the same limit. [Note: The common limit of the sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$ is called the arithmetic-geometric mean of the nonnegative real numbers $\alpha$ and $\beta$. It was introduced and studied by Gauss. For further details, see [20].]

M Z

Numerade Educator

Problem 13

If a monotonic sequence $\left(a_{n}\right)$ has a subsequence $\left(a_{n_{k}}\right)$ such that $a_{n_{k}} \rightarrow a$ where $a \in \mathbb{R}$ or $a=\infty$ or $a=-\infty$, then show that $a_{n} \rightarrow a$.

Vishnu P

Numerade Educator

Problem 14

Prove that a sequence $\left(a_{n}\right)$ in $\mathbb{R}$ has no convergent subsequence if and only if $\left|a_{n}\right| \rightarrow \infty$.

Vishnu P

Numerade Educator

Problem 15

Let $\left(a_{n}\right)$ be a sequence of real numbers and let $a \in \mathbb{R} .$ Show that $a_{n} \rightarrow a$ if and only if every subsequence of $\left(a_{n}\right)$ has a subsequence converging to a. (Hint: Proposition 2.17.)

Vishnu P

Numerade Educator

Problem 16

A real number $a$ is called a cluster point of a sequence $\left(a_{n}\right)$ in $\mathbb{R}$ if there is a subsequence $\left(a_{n_{k}}\right)$ of $\left(a_{n}\right)$ such that $a_{n_{k}} \rightarrow a$
(i) Show that if $a_{n} \rightarrow a$, then $a$ is the only cluster point of $\left(a_{n}\right)$.
(ii) Show that the converse of (i) is not true. In other words, show that there is a divergent sequence that has a unique cluster point. (Hint:
$a_{2 k}:=\frac{1}{2 k}$ and $a_{2 k+1}:=2 k+1$ for $k \in \mathbb{N}$.)
(iii) Show that if $a_{n} \rightarrow \infty$ or if $a_{n} \rightarrow-\infty$, then $\left(a_{n}\right)$ has no cluster point.
(iv) Show that the converse of (iii) is not true. In other words, show that there is a sequence without a cluster point that neither tends to $\infty$ nor tends to $-\infty .$ (Hint: $a_{n}:=(-1)^{n} n$ for $n \in \mathbb{N}$.)

Faizanullah Kazmi

Faizanullah Kazmi

Numerade Educator

Problem 17

Let $A_{n}:=1+(1 / 2)+\cdots+(1 / n)$ for $n \in \mathbb{N}$. Show that $\left(A_{n+1}-A_{n}\right) \rightarrow 0$ as $n \rightarrow \infty$, but $\left(A_{n}\right)$ is not a Cauchy sequence.

Nick Johnson

Numerade Educator

Problem 18

Let $A_{n}:=1+\left(1 / 2^{2}\right)+\cdots+\left(1 / n^{2}\right)$ for $n \in \mathbb{N}$. Show that there is no real number $\alpha<1$ such that $\left|A_{n+1}-A_{n}\right| \leq \alpha\left|A_{n}-A_{n-1}\right|$ for all $n \in \mathbb{N}$ with $n \geq 2$, but $\left(A_{n}\right)$ is a Cauchy sequence.

Nick Johnson

Numerade Educator

Problem 19

Let $x \in \mathbb{R}$ and $x>0$. Define
$A_{n}:=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\cdots+\frac{x^{n}}{n !} \quad$ and $\quad B_{n}:=\left(1+\frac{x}{n}\right)^{n} \quad$ for $n \in \mathbb{N} .$
Show that $\left(A_{n}\right)$ and $\left(B_{n}\right)$ are convergent and have the same limit.

Linh Vu

Numerade Educator

Problem 20

Show that the number $e:=\lim _{n \rightarrow \infty} \sum_{k=0}^{n}(1 / k !)$ is irrational. (Hint: For every $n \in \mathbb{N}, 0<e \sum_{k=0}^{n}(1 / k !)<(1 / n ! n)$. Multiply by $n !$.)

Wendi Zhao

Numerade Educator

Problem 21

(i) If $\left(a_{n}\right)$ is a sequence and $a_{n} \rightarrow a$, then show that $\left(a_{1}+\cdots+a_{n}\right) / n \rightarrow a$. Here $a \in \mathbb{R}$ or $a=\infty$ or $a=-\infty .$ Give an example to show that the converse is not true.
Exercises
(ii) Find $\lim _{n \rightarrow \infty} \frac{1}{n}\left(\frac{2}{5}+\frac{5}{11}+\cdots+\frac{n^{2}+1}{2 n^{2}+3}\right)$.

Mengchun Cai

Numerade Educator

Problem 22

Suppose $\alpha, \beta$, and $\gamma$ are positive real numbers. Let
$$
a_{1}:=\alpha \quad \text { and } \quad a_{n+1}:=\frac{a_{n}}{\beta a_{n}+\gamma} \quad \text { for } n \in \mathbb{N} .
$$
Show that $\left(a_{n}\right)$ is convergent. Further, if $a:=\lim _{n \rightarrow \infty} a_{n}$, then show that $a=0$ if $\gamma \geq 1$ and $a=(1-\gamma) / \beta$ otherwise. (Hint: Consider the cases $\alpha \beta+\gamma \geq 1$ and $\alpha \beta+\gamma<1 .)$

Mengchun Cai

Numerade Educator

Problem 23

Suppose $\alpha$ and $\beta$ are nonnegative real numbers. Let
$$
a_{1}:=\alpha \quad \text { and } \quad a_{n+1}:=\sqrt{\beta+a_{n}} \quad \text { for } n \in \mathbb{N}
$$
Show that $\left(a_{n}\right)$ is convergent. Further, if $a:=\lim _{n \rightarrow \infty} a_{n}$, then show that $a=0$ if $\alpha=0=\beta$, and $a=(1+\sqrt{1+4 \beta}) / 2$ otherwise. (Hint: Consider the cases $\sqrt{\alpha+\beta} \leq \alpha$ and $\sqrt{\alpha+\beta}>\alpha$.)

Nick Johnson

Numerade Educator

Problem 24

Suppose $\alpha$ and $\beta$ are nonnegative real numbers. Let
$$
a_{1}:=\alpha \quad \text { and } \quad a_{n+1}:=\beta+\sqrt{a_{n}} \quad \text { for } n \in \mathbb{N}
$$
Show that $\left(a_{n}\right)$ is convergent. Further, if $a:=\lim _{n \rightarrow \infty} a_{n}$, then show that $a=0$ if $\alpha=0=\beta$, and $a=(1+2 \beta+\sqrt{1+4 \beta}) / 2$ otherwise. (Hint:
Consider the cases $\sqrt{\alpha}+\beta \leq \alpha$ and $\sqrt{\alpha}+\beta>\alpha$.)

Nick Johnson

Numerade Educator

Problem 25

Let $\left(a_{n}\right)$ and $\left(b_{n}\right)$ be sequences such that $\left|a_{n+1}-a_{n}\right| \leq b_{n}$ for all $n \in \mathbb{N}$. Define
$$
B_{n}:=\sum_{k=1}^{n} b_{k} \quad \text { for } n \in \mathbb{N} .
$$
If $\left(B_{n}\right)$ is convergent, then show that $\left(a_{n}\right)$ is a Cauchy sequence and hence it is convergent.

Nick Johnson

Numerade Educator

Problem 26

Let $y$ be any real number with $0 \leq y<1$. Define sequences $\left(b_{n}\right)$ and $\left(y_{n}\right)$ iteratively as follows. Let $y_{1}:=10 y$ and $b_{1}:=\left[y_{1}\right]$, and for each $n \in \mathbb{N}$,
$$
y_{n+1}:=10\left(y_{n}-b_{n}\right) \quad \text { and } \quad b_{n+1}:=\left[y_{n+1}\right] \text { . }
$$
Show that for each $n \in \mathbb{N}$ we have
$$
0 \leq y_{n}<10 \quad \text { and } \quad b_{n} \in \mathbb{Z} \text { with } 0 \leq b_{n} \leq 9
$$
and moreover,
$$
y=\frac{b_{1}}{10}+\frac{b_{2}}{10^{2}}+\cdots+\frac{b_{n}}{10^{n}}+\frac{y_{n+1}}{10^{n+1}}
$$
Deduce that
$$
0 \leq \frac{y_{n+1}}{10^{n+1}}<\frac{1}{10^{n}} \quad \text { for each } n \in \mathbb{N}
$$
and consequently,
$$
y=\lim _{n \rightarrow \infty}\left(\frac{b_{1}}{10}+\frac{b_{2}}{10^{2}}+\cdots+\frac{b_{n}}{10^{n}}\right).
$$
[Note: It is customary to call the nonnegative integers $b_{1}, b_{2}, \ldots$, the digits of $y$ and write the above expression for $y$ as $y=0 . b_{1} b_{2} \ldots$, and call it the decimal expansion of $y$.]

Mengchun Cai

Numerade Educator

Problem 27

Given any $m \in \mathbb{N}$, show that there is a unique nonnegative integer $k$ such that $10^{k} \leq m<10^{k+1}$. Use Exercise 37 of Chapter 1 repeatedly to show that there are unique integers $a_{0}, a_{1}, \ldots, a_{k}$ between 0 and 9 such that
$$
m=a_{0}+a_{1}(10)+a_{2}\left(10^{2}\right)+\cdots+a_{k}\left(10^{k}\right) \text { . }
$$

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 28

Given any $x \in \mathbb{R}$, show that there is a nonnegative integer $k$ and integers $a_{k}, a_{k-1}, \ldots, a_{1}, a_{0}, b_{1}, b_{2}, \ldots$ between 0 and 9 such that
$$
x=\pm \lim _{n \rightarrow \infty}\left(a_{k} 10^{k}+a_{k-1} 10^{k-1}+\cdots+a_{0}+\frac{b_{1}}{10}+\frac{b_{2}}{10^{2}}+\cdots+\frac{b_{n}}{10^{n}}\right) .
$$
(Hint: If $|x|<1$, set $k=0=a_{0}$ and apply Exercise 26 to $y:=|x|$, whereas if $|x| \geq 1$, apply Exercise 27 to $n:=[|x|]$ and Exercise 26 to $y:=|x|-n .)$ [Note: It is customary to call $a_{k}, a_{k-1}, \ldots, a_{0}, b_{1}, b_{2}, \ldots$ the digits of $x$ and write the above expression for $x$ as $x=\pm a_{k} a_{k-1} \ldots a_{0} \cdot b_{1} b_{2} \ldots$, and call it the decimal expansion of $x$.

Colin O'Haire

Numerade Educator

Problem 29

Given any $y \in[0,1)$, let $\left(y_{n}\right)$ and $\left(b_{n}\right)$ be the sequences associated to $y$ as in Exercise 26 . We say that the decimal expansion of $y$ is finite if $y_{n}=0$ for some $n \in \mathbb{N}$ and recurring if it not finite but $y_{i}=y_{j}$ for some $i, j \in \mathbb{N}$ with $i<j$. Show that if $y \in[0,1)$ is a rational number, then its decimal expansion is either finite or recurring. (Hint: Write $y$ in reduced form as $y=p / q$. Let $r_{0}:=p$. Use Exercise 37 of Chapter 1 successively to find integers $q_{1}, r_{1}, q_{2}, r_{2}, \ldots$ such that $10 r_{i-1}=q q_{i}+r_{i}$ and $0 \leq r_{i}<q$ for $i \geq 1$. Now $y_{i}=10 r_{i-1} / q$ and the $r_{i}$ 's take only finitely many values. [Note: The converse also holds. see Remark $9.2$.

Geena Pullo

Numerade Educator

Problem 30

Show that the results of Exercises $26,27,28$, and 29 are valid with the number 10 replaced by any integer $d>1$ and the number 9 by $d-1$. [Note: The corresponding limiting expression of a real number $x$ is called the $d$ -ary expansion of $x$. When $d=2$, it is called the binary expansion and when $d=3$, it is called the ternary expansion.]

James Chok

Numerade Educator

Problem 31

Define
$$
a_{1}:=1 \quad \text { and } \quad a_{n+1}:=\left(1+\frac{(-1)^{n}}{2^{n}}\right) a_{n} \quad \text { for } n \in \mathbb{N} \text { . }
$$
(i) For every $n \in \mathbb{N}$, show that
$$
\left|a_{n+1}\right| \leq\left(1+\frac{1}{2^{n}}\right)\left(1+\frac{1}{2^{n-1}}\right) \cdots\left(1+\frac{1}{2}\right) \leq\left(\frac{n+1}{n}\right)^{n}<3
$$
(Hint: Use the A.M.-G.M. inequality.)
Exercises
(ii) Use (i) above to show that
$$
\left|a_{n+1}-a_{n}\right|<\frac{3}{2^{n}} \quad \text { for all } n \in \mathbb{N}
$$
Deduce, using Exercise 25 , that $\left(a_{n}\right)$ is a Cauchy sequence.
(iii) Conclude that $\left(a_{n}\right)$ is convergent. Is $\left(a_{n}\right)$ monotonic?

Chris Trentman

Numerade Educator

Problem 32

Assuming only the algebraic and the order properties of $\mathbb{R}$, and assuming that every monotonically decreasing sequence that is bounded below is convergent in $\mathbb{R}$, establish the Completeness Property of $\mathbb{R}$. (Hint: Consider $S \subseteq \mathbb{R}, a_{0} \in S$, and an upper bound $\alpha_{0}$ of $S .$ If $\left(a_{0}+\alpha_{0}\right) / 2$ is an upper bound of $S$, let $a_{1}:=a_{0}$ and $\alpha_{1}:=\left(a_{0}+\alpha_{0}\right) / 2 ;$ otherwise, there is $a_{1} \in S$ such that $\left(a_{0}+\alpha_{0}\right) / 2<a_{1}$ and in this case, let $\alpha_{1}:=\alpha_{0} .$ Continuing in this manner, obtain a monotonically decreasing sequence $\left(\alpha_{n}\right)$ that is bounded below.) [Compare part (ii) of Proposition 2.8.]

Nick Johnson

Numerade Educator

Problem 33

(Nested Interval Theorem) Let $I_{n}:=\left[a_{n}, b_{n}\right], n \in \mathbb{N}$, be closed intervals such that $I_{n} \supseteq I_{n+1}$ for all $n \in \mathbb{N}$ and $\left|b_{n}-a_{n}\right| \rightarrow 0 .$ Show that there is a unique $x \in \mathbb{R}$ such that $x \in I_{n}$ for all $n \in \mathbb{N}$. (Hint: Use Exercise 51 of Chapter $1 .$ )

Stanley Enemuo

Numerade Educator

Problem 34

Use the Nested Interval Theorem in Exercise 33 to prove the BolzanoWeierstrass Theorem.

Nick Johnson

Numerade Educator

Problem 35

Let $\left(a_{n}\right)$ be a sequence in $\mathbb{R}$.
(i) Assume that $\left(a_{n}\right)$ is bounded above and $a_{n} \not \rightarrow-\infty$. Define
$M_{n}:=\sup \left\{a_{n}, a_{n+1}, \ldots\right\}$ for $n \in \mathbb{N} \quad$ and $\quad M:=\inf \left\{M_{1}, M_{2}, \ldots\right\}$
Show that the sequence $\left(M_{n}\right)$ converges to $M$ and $M$ is the largest cluster point of $\left(a_{n}\right)$.
(ii) Assume that $\left(a_{n}\right)$ is bounded below and $a_{n} \not \leftrightarrow \infty$. Define
$m_{n}:=\inf \left\{a_{n}, a_{n+1}, \ldots\right\}$ for $n \in \mathbb{N} \quad$ and $\quad m:=\sup \left\{m_{1}, m_{2}, \ldots\right\}$
Show that the sequence $\left(m_{n}\right)$ converges to $m$ and $m$ is the smallest cluster point of $\left(a_{n}\right)$. [See Exercise 16 for the definition of a cluster point.]

Vishnu P

Numerade Educator

Problem 36

Let $\left(a_{n}\right)$ be a sequence in $\mathbb{R}$. Define the limit superior (or the upper limit) of $\left(a_{n}\right)$ by
$$
\limsup _{n \rightarrow \infty} a_{n}:=\left\{\begin{array}{ll}
\lim _{n \rightarrow \infty} M_{n} & \text { if }\left(a_{n}\right) \text { is bounded above and } a_{n} \not \rightarrow-\infty \\
\infty & \text { if }\left(a_{n}\right) \text { is not bounded above } \\
-\infty & \text { if } a_{n} \rightarrow-\infty
\end{array}\right.
$$
where the sequence $\left(M_{n}\right)$ is as defined in Exercise 35 . Similarly, define the limit inferior (or the lower limit) of $\left(a_{n}\right)$ by
$$
\liminf _{n \rightarrow \infty} a_{n}:=\left\{\begin{array}{ll}
\lim _{n \rightarrow \infty} m_{n} & \text { if }\left(a_{n}\right) \text { is bounded below and } a_{n} \rightarrow \infty \\
-\infty & \text { if }\left(a_{n}\right) \text { is not bounded below, } \\
\infty & \text { if } a_{n} \rightarrow \infty
\end{array}\right.
$$
where the sequence $\left(m_{n}\right)$ is as defined in Exercise 35 . If $\left(a_{n}\right)$ is bounded, then show that the set $C$ of all cluster points of $\left(a_{n}\right)$ is nonempty, and moreover,
$$
\limsup _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} \sup \left\{a_{n}, a_{n+1}, \ldots\right\}=\max C
$$
and
$$
\liminf _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} \inf \left\{a_{n}, a_{n+1}, \ldots\right\}=\min C.
$$

Linh Vu

Numerade Educator

Problem 37

Determine $\lim \sup _{n \rightarrow \infty} a_{n}$ and $\lim \inf _{n \rightarrow \infty} a_{n}$ if $\left(a_{n}\right)$ is as defined below.
(i) $a_{n}:=(-1)^{n}\left(1+\frac{1}{n}\right)$ for $n \in \mathbb{N}$,
(ii) $a_{n}:=(-1)^{n} n$ for $n \in \mathbb{N}$,
(iii) $a_{1}:=0$ and for $k \in \mathbb{N}, a_{2 k}:=a_{2 k-1} / 2$ and $a_{2 k+1}:=(1 / 2)+a_{2 k}$. (Hint: $a_{2 k}=(1 / 2)-\left(1 / 2^{k}\right)$ for all $\left.k \in \mathbb{N} .\right)$

Lucas Finney

Numerade Educator

Problem 38

Let $\left(r_{n}\right)$ be a sequence such that $\mathbb{Q}=\left\{r_{n}: n \in \mathbb{N}\right\} .$ [Note that by Exercise 49 (iii) of Chapter 1, such a sequence exists.] Determine the set of all cluster points of $\left(r_{n}\right)$, and also $\lim \inf _{n \rightarrow \infty} r_{n}$ as well as $\lim \sup _{n \rightarrow \infty} r_{n}$.

Nick Johnson

Numerade Educator

Problem 39

Let $\left(a_{n}\right)$ be a sequence in $\mathbb{R}$. Prove the following:
(i) $\lim \inf _{n \rightarrow \infty} a_{n} \leq \lim \sup _{n \rightarrow \infty} a_{n}$.
(ii) $\left(a_{n}\right)$ is bounded if and only if both $\lim \inf _{n \rightarrow \infty} a_{n}$ and $\lim \sup _{n \rightarrow \infty} a_{n}$
are real numbers.
(iii) $\left(a_{n}\right)$ is convergent if and only if both $\lim \inf _{n \rightarrow \infty} a_{n}$ and $\lim \sup _{n \rightarrow \infty} a_{n}$
are real numbers and are equal to each other. In this case,
$$
\liminf _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} a_{n}=\limsup _{n \rightarrow \infty} a_{n}
$$
(iv) $a_{n} \rightarrow \infty$ if and only if $\lim \inf _{n \rightarrow \infty} a_{n}=\infty=\lim \sup _{n \rightarrow \infty} a_{n}$.
(v) $a_{n} \rightarrow-\infty$ if and only if $\lim \inf _{n \rightarrow \infty} a_{n}=-\infty=\limsup _{n \rightarrow \infty} a_{n}$.

Linh Vu

Numerade Educator

Problem 40

Let $\left(a_{n}\right)$ be a sequence in $\mathbb{R}$. Prove Corollary $2.16$ (which is a more elaborate version of the Bolzano-Weierstrass Theorem) by showing that if $\left(a_{n}\right)$ is bounded above and $a_{n} \not \rightarrow-\infty$, then $\left(a_{n}\right)$ has a subsequence that converges to $\lim \sup _{n \rightarrow \infty} a_{n}$, while if $\left(a_{n}\right)$ is bounded below and $a_{n} \not \leftrightarrow \infty$
then $\left(a_{n}\right)$ has a subsequence that converges to $\lim \inf _{n \rightarrow \infty} a_{n}$.

Vishnu P

Numerade Educator

Problem 41

Let $\left(a_{n}\right)$ be a Cauchy sequence in $\mathbb{R} .$ Prove that $\left(a_{n}\right)$ is convergent by showing that it is bounded and $\lim \sup _{n \rightarrow \infty} a_{n}=\lim \inf _{n \rightarrow \infty} a_{n}$.

Vishnu P

Numerade Educator

Problem 42

Assuming only the algebraic and the order properties of $\mathbb{R}$, and assuming that every Cauchy sequence in $\mathbb{R}$ is convergent, establish the Completeness Property of $\mathbb{R}$. (Hint: Consider $S \subseteq \mathbb{R}$ and $a_{n}, \alpha_{n}$ as in the Hint for Exercise 32. Then $\alpha_{n}-a_{n} \leq\left(\alpha_{0}-a_{0}\right) / 2^{n}$ for all $n \in \mathbb{N}$.)

Nick Johnson

Numerade Educator