Real analysis

N. L. Carothers

Chapter 1 Calculus Review - all with Video Answers

Educators

Chapter Questions

Problem 1

If $A$ is a nonempty subset of $\mathbb{R}$ that is bounded below, show that $A$ has a greatest lower bound. That is, show that there is a number $m \in \mathbb{R}$ satisfying: (i) $m$ is a lower bound for $A$; and (ii) if $x$ is a lower bound for $A$, then $x \leq m$. [Hint: Consider the set $-A=\{-a: a \in A\}$ and show that $m=-\sup (-A)$ works.]

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Problem 2

Let $A$ be a bounded subset of $\mathbb{R}$ containing at least two points. Prove:
(a) $-\infty<\inf A<\sup A<+\infty$.
(b) If $B$ is a nonempty subset of $A$, then $\inf A \leq \inf B \leq \sup B \leq \sup A$.
(c) If $B$ is the set of all upper bounds for $A$, then $B$ is nonempty, bounded below, and $\inf B=\sup A$.

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Problem 3

Establish the following apparently different (but "fancier") characterization of the supremum. Let $A$ be a nonempty subset of $\mathbf{R}$ that is bounded above. Prove that $s=\sup A$ if and only if (i) $s$ is an upper bound for $A$, and (ii) for every $\varepsilon>0$, there is an $a \in A$ such that $a>s-\varepsilon$. State and prove the corresponding result for the infimum of a nonempty subset of $\mathbb{R}$ that is bounded below.
Recall that a sequence ( $x_n$ ) of real numbers is said to converge to $x \in \mathbb{R}$ if, for every $\varepsilon>0$, there is a positive integer $N$ such that $\left|x_n-x\right|<\varepsilon$ whenever $n \geq N$. In this case, we call $x$ the limit of the sequence $\left(x_n\right)$ and write $x=\lim _{n \rightarrow \infty} x_n$.

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Problem 4

Let $A$ be a nonempty subset of $\mathbf{R}$ that is bounded above. Show that there is a sequence $\left(x_n\right)$ of elements of $A$ that converges to $\sup A$.

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01:49

Problem 5

Suppose that $a_n \leq b$, for all $n$, and that $a=\lim _{n \rightarrow \infty} a_n$ exists. Show that $a \leq b$. Conclude that $a \leq \sup _n a_n=\sup \left\{a_n: n \in \mathbb{N}\right\}$.

Nick Johnson

Numerade Educator

07:49

Problem 6

Prove that every convergent sequence of real numbers is bounded. Moreover, if ( $a_n$ ) is convergent, show that $\inf _n a_n \leq \lim _{n \rightarrow \infty} a_n \leq \sup _n a_n$.

Muhammad Saleem

Numerade Educator

02:06

Problem 7

If $a<b$, then there is also an irrational $x \in \mathbb{R} \backslash \mathbb{Q}$ with $a<x<b$. [Hint: Find an irrational of the form $p \sqrt{2} / q$.]

Adriano Chikande

Numerade Educator

01:23

Problem 8

Given $a<b$, show that there are, in fact, infinitely many distinct rationals between $a$ and $b$. The same goes for irrationals, too.

Carson Merrill

Numerade Educator

01:14

Problem 9

Show that the least upper bound axiom also holds in $\mathbb{Z}$ (i.e., each nonempty subset of $\mathbb{Z}$ with an upper bound in $\mathbb{Z}$ has a least upper bound in $\mathbb{Z}$ ), but that it fails to hold in $\mathbb{Q}$.

Faizanullah Kazmi

Numerade Educator

01:36

Problem 10

Let $a_1=\sqrt{2}$ and let $a_{n+1}=\sqrt{2 a_n}$ for $n \geq 1$. Show that $\left(a_n\right)$ converges and find its limit. [Hint: Show that $\left(a_n\right)$ is increasing and bounded.]

Nick Johnson

Numerade Educator

Problem 11

Fix $a>0$ and let $x_1>\sqrt{a}$. For $n \geq 1$, define
$$
x_{n+1}=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right)
$$

Show that $\left(x_n\right)$ converges and that $\lim _{n \rightarrow \infty} x_n=\sqrt{a}$.

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00:27

Problem 12

Suppose that $s_1>s_2>0$ and let $s_{n+1}=\frac{1}{2}\left(s_n+s_{n-1}\right)$ for $n \geq 2$. Show that $\left(s_n\right)$ converges. [Hint: Show that $\left(s_{2 n-1}\right)$ decreases and $\left(s_{2 n}\right)$ increases.]

Sahil Patel

Numerade Educator

02:56

Problem 13

Let $a_n \geq 0$ for all $n$, and let $s_n=\sum_{i=1}^n a_i$. Show that $\left(s_n\right)$ converges if and only if $\left(s_n\right)$ is bounded.

Recall that a sequence of real numbers $\left(x_n\right)$ is said to be Cauchy if, for every $\varepsilon>0$, there is an integer $N \geq 1$ such that $\left|x_n-x_m\right|<\varepsilon$ whenever $n, m \geq N$.

Nick Johnson

Numerade Educator

07:49

Problem 14

Prove that a convergent sequence is Cauchy, and that any Cauchy sequence is bounded.

Muhammad Saleem

Numerade Educator

Problem 15

Show that a Cauchy sequence with a convergent subsequence actually converges.

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05:55

Problem 16

(a) Why is $0.4999 \ldots=0.5$ ? (Try to give more than one reason.)
(b) Write $0.234234234 \ldots$ as a fraction.
(c) Precisely which real numbers between 0 and 1 have more than one decimal representation? Explain.

J Hardin

Numerade Educator

02:25

Problem 17

Given real numbers $a$ and $b$, establish the following formulas: $|a+b| \leq$ $|a|+|b|,|| a|-| b|| \leq|a-b|, \max \{a, b\}=\frac{1}{2}(a+b+|a-b|)$, and $\min \{a, b\}=$ $\frac{1}{2}(a+b-|a-b|)$.

Teresa Fuston

Numerade Educator

02:19

Problem 18

(a) Given $a>-1, a \neq 0$, use induction to show that $(1+a)^n>1+n a$ for any integer $n>1$.
(b) Use (a) to show that, for any $x>0$, the sequence $(1+(x / n))^n$ increases.
(c) If $a>0$, show that $(1+a)^r>1+r a$ holds for any rational exponent $r>1$.
[Hint: If $r=p / q$, then apply (a) with $n=q$ and (b) with $x=a p$.]
(d) Finally, show that (c) holds for any real exponent $r>1$.

Sanchit Gogia

Numerade Educator

01:35

Problem 19

If $0<c<1$, show that $c^n \rightarrow 0$; and if $c>0$, show that $c^{1 / n} \rightarrow 1$. [Hint: Use Bernoulli's inequality for each, once with $c=1 /(1+x), x>0$ and once with $c^{1 / n}=1+x_n$, where $\left.x_n>0.\right]$

Nick Johnson

Numerade Educator

01:30

Problem 20

Given $a, b>0$, show that $\sqrt{a b} \leq \frac{1}{2}(a+b)$ (this is the arithmetic-geometric mean inequality). Generalize this to $\left(a_1 \cdot a_2 \cdots a_n\right)^{1 / n} \leq(1 / n)\left(a_1+a_2+\cdots+a_n\right)$. [Hint: Induction and Bernoulli's inequality.]

Carson Merrill

Numerade Educator

15:48

Problem 21

Let $p \geq 2$ be a fixed integer, and let $0<x<1$. If $x$ has a finite-length base $p$ decimal expansion, that is, if $x=a_1 / p+\cdots+a_n / p^n$ with $a_n \neq 0$. prove that $x$ has precisely two base $p$ decimal expansions. Otherwise, show that the base $p$ decimal expansion for $x$ is unique. Characterize the numbers $0<x<1$ that have repeating base $p$ decimal expansions. How about eventually repeating?

Geena Pullo

Numerade Educator

Problem 22

Show that $\inf _n a_n \leq \liminf _{n \rightarrow \infty} a_n \leq \limsup _{n \rightarrow \infty} a_n \leq \sup _n a_n$.

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Problem 23

If $\left(a_n\right)$ is convergent, show that $\liminf \operatorname{inc}_n a_n=\limsup \sin _{n \rightarrow \infty} a_n=\lim _{n \rightarrow \infty} a_n$.

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Problem 24

Show that $\lim \sup _{n \rightarrow \infty}\left(-a_n\right)=-\liminf _{n \rightarrow \infty} a_n$.

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Problem 25

If $\lim \sup _{n \rightarrow \infty} a_n=-\infty$, show that $\left(a_n\right)$ diverges to $-\infty$. If $\lim \sup _{n \rightarrow \infty} a_n=$ $+\infty$, show that $\left(a_n\right)$ has a subsequence that diverges to $+\infty$. What happens if $\liminf _{n \rightarrow \infty} a_n= \pm \infty$ ?

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Problem 26

Prove the characterization of lim sup given above. That is, given a bounded sequence $\left(a_n\right)$, show that the number $M=\lim \sup _{n \rightarrow \infty} a_n$ satisfies ( $*$ ) and, conversely, that any number $M$ satisfying (*) must equal $\lim \sup _{n \rightarrow \infty} a_n$. State and prove the corresponding result for $m=\liminf _{n \rightarrow \infty} a_n$.

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Problem 27

Prove that every sequence of real numbers $\left(a_n\right)$ has a subsequence $\left(a_{n_k}\right)$ that converges to lim sup $n_{n \rightarrow \infty} a_n$. [Hint: If $M=\lim \sup _{n \rightarrow \infty} a_n= \pm \infty$, we must interpret the conclusion loosely; this case is handled in Exercise 25. If $M \neq \pm \infty$, use (*) to choose ( $a_{n_k}$ ) satisfying $\left|a_{n_k}-M\right|<1 / k$, for example.] There is necessarily also a subsequence that converges to liminf ${ }_{n \rightarrow \infty} a_n$. Why?

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Problem 28

By modifying the argument in the previous exercise, show that every sequence of real numbers has a monotone subsequence.

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Problem 29

If $\left(a_{n_k}\right)$ is a convergent subsequence of $\left(a_n\right)$, show that $\liminf _{n \rightarrow \infty} a_n \leq$ $\lim _{k \rightarrow \infty} a_{n_k} \leq \lim \sup _{n \rightarrow \infty} a_n$.

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01:49

Problem 30

If $a_n \leq b_n$ for all $n$, and if $\left(a_n\right)$ converges, show that $\lim _{n \rightarrow \infty} a_n \leq \liminf _{n \rightarrow \infty}$ $b_n$.

Nick Johnson

Numerade Educator

Problem 31

If $\left(a_n\right)$ is convergent and $\left(b_n\right)$ is bounded, show that lim sup ${ }_{n \rightarrow \infty}\left(a_n+b_n\right) \leq$ $\lim _{n \rightarrow \infty} a_n+\lim \sup _{n \rightarrow \infty} b_n$.

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04:38

Problem 32

Given a sequence $\left(a_n\right)$ of real numbers, let $\mathcal{S}$ be the set of all limits of convergent subsequences of $\left(a_n\right)$ (including, possibly, $\pm \infty$ ). For example, it follows from Exercise 27 that $\lim \sup _{n \rightarrow \infty} a_n$ and $\lim _{n \rightarrow \infty} a_n$ are both elements of $\mathcal{S}$. Show that, in fact, $\lim \sup _{n \rightarrow \infty} a_n=\sup \mathcal{S}$ and $\liminf _{n \rightarrow \infty} a_n=\inf \mathcal{S}$.

Shu-Ting Huang

Numerade Educator

06:50

Problem 33

Show that $\left(x_n\right)$ converges to $x \in \mathbb{R}$ if and only if every subsequence $\left(x_{n_k}\right)$ of $\left(x_n\right)$ has a further subsequence ( $x_{n_{k_1}}$ ) that converges to $x$.

Supratim Pal

Numerade Educator

Problem 34

Suppose that $a_n \geq 0$ and that $\sum_{n=1}^{\infty} a_n<\infty$.
(i) Show that $\liminf _{n \rightarrow \infty} n a_n=0$.
(ii) Give an example showing that $\lim \sup _{n \rightarrow \infty} n a_n>0$ is possible.

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06:35

Problem 35

(The ratio test): Let $a_n \geq 0$.
(i) If $\limsup \operatorname{sum}_{n \rightarrow \infty} a_{n+1} / a_n<1$, show that $\sum_{n=1}^{\infty} a_n<\infty$.
(ii) If lim inf ${ }_{n \rightarrow \infty} a_{n+1} / a_n>1$, show that $\sum_{n=1}^{\infty} a_n$ diverges.
(iii) Find examples of both a convergent and a divergent series having $\lim _{n \rightarrow \infty} a_{n+1} / a_n=1$.

Maya Mysore

Numerade Educator

Problem 36

(The root test): Let $a_n \geq 0$.
(i) If lim $\sup _{n \rightarrow \infty} \sqrt[n]{a_n}<1$, show that $\sum_{n=1}^{\infty} a_n<\infty$.
(iii) Find examples of both a convergent and a divergent series having $\lim _{n \rightarrow \infty} \sqrt[n]{a_n}=1$

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Problem 37

If $\left(E_n\right)$ is a sequence of subsets of a fixed set $S$, we define
$$
\underset{n \rightarrow \infty}{\limsup } E_n=\bigcap_{n=1}^{\infty}\left(\bigcup_{k=n}^{\infty} E_k\right) \text { and } \liminf _{n \rightarrow \infty} E_n=\bigcup_{n=1}^{\infty}\left(\bigcap_{k=n}^{\infty} E_k\right) \text {. }
$$

Show that
$$
\liminf _{n \rightarrow \infty} E_n \subset \underset{n \rightarrow \infty}{\limsup } E_n \text { and that } \underset{n \rightarrow \infty}{\liminf }\left(E_n^c\right)=\left(\limsup _{n \rightarrow \infty} E_n\right)^c \text {. }
$$

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02:29

Problem 38

Show that
$$
\underset{n \rightarrow \infty}{\limsup } E_n=\left\{x \in S: x \in E_n \text { for infinitely many } n\right\}
$$
and that
$$
\liminf _{n \rightarrow \infty} E_n=\left\{x \in S: x \in E_n \text { for all but finitely many } n\right\} .
$$

Andreas Papavassiliou

Numerade Educator

Problem 39

How would you define the limit (if it exists) of a sequence of sets? What should the limit be if $E_1 \supset E_2 \supset \cdots$ ? If $E_1 \subset E_2 \subset \cdots$ ? Compute $\liminf _{n \rightarrow \infty} E_n$ and $\lim \sup _{n \rightarrow \infty} E_n$ in both cases and test your conjecture.

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Problem 40

Prove Theorem 1.17.

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03:37

Problem 41

Prove Theorem 1.18 , including 1.18 (iv) as one of the equivalent conditions.

Wendi Zhao

Numerade Educator

01:40

Problem 42

Given $f:(a, b) \rightarrow \mathbb{R}$ and $x \in(a, b)$, consider the statements: (i) $\lim _{h \rightarrow 0}$ $|f(x+h)-f(x)|=0$ and (ii) $\lim _{h \rightarrow 0}|f(x+h)-f(x-h)|=0$. Show that (i) always implies (ii). Give an example where (ii) holds but not (i).

Andreas Papavassiliou

Numerade Educator

02:37

Problem 43

Modify Theorem 1.17 to characterize the statement $\lim _{x \rightarrow a^{+}} f(x)=L$, and check your new version by providing a proof!

Nick Johnson

Numerade Educator

Problem 44

If $f: \mathbb{R} \rightarrow \mathbb{R}$ is increasing and bounded, show that $\lim _{x \rightarrow \infty} f(x)$ and $\lim _{x \rightarrow-\infty} f(x)$ both exist.

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Problem 45

Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous and suppose that $f(x)=0$ whenever $x$ is rational. Show that $f(x)=0$ for every $x$ in $[a, b]$.

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01:03

Problem 46

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous.
(a) If $f(0)>0$, show that $f(x)>0$ for all $x$ in some open interval $(-a, a)$.
(b) If $f(x) \geq 0$ for every rational $x$, show that $f(x) \geq 0$ for all real $x$. Will this result hold with " $\geq 0$ " replaced by " $>0$ "? Explain.

Carson Merrill

Numerade Educator

38:45

Problem 47

Let $f, g, h$, and $k$ be defined on $[0,1]$ as follows:
$$
\begin{array}{ll}
f(x)= \begin{cases}0 & \text { if } x \notin \mathbb{Q} \\
1 & \text { if } x \in \mathbb{Q}\end{cases} & h(x)= \begin{cases}1-x & \text { if } x \notin \mathbb{Q} \\
x & \text { if } x \in \mathbb{Q}\end{cases} \\
g(x)= \begin{cases}0 & \text { if } x \notin \mathbb{Q} \\
x & \text { if } x \in \mathbb{Q}\end{cases} & k(x)= \begin{cases}0 & \text { if } x \notin \mathbb{Q} \\
1 / n & \text { if } x=m / n \in \mathbb{Q} \\
& \text { (in lowest terms). }\end{cases}
\end{array}
$$
Prove that $f$ is not continuous at any point in $[0,1]$, that $g$ is continuous only at $x=0$, that $h$ is continuous only at $x=1 / 2$, and that $k$ is continuous only at the irrational points in $[0,1]$.

Chris Trentman

Numerade Educator

Problem 48

Give an example of a one-to-one, onto function $f:[0,1] \rightarrow[0,1]$ that is not monotone. Can you find a monotone, one-to-one function that is not onto? Or a monotone, onto function that is not one-to-one?

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01:13

Problem 49

Let $f:(a, b) \rightarrow \mathbb{R}$ be monotone and let $a<x<b$. Show that $f$ is continuous at $x$ if and only if $f(x-)=f(x+)$.

Carson Merrill

Numerade Educator

Problem 50

Let $D$ denote the set of rationals in $[0,1]$ and suppose that $f: D \rightarrow \mathbb{R}$ is increasing. Show that there is an increasing function $g:[0,1] \rightarrow \mathbb{R}$ such that $g(x)=f(x)$ whenever $x$ is rational. [Hint: For $x \in[0,1]$, define $g(x)=\sup (f(t)$ : $0 \leq t \leq 1, t \in \mathbb{Q}\}$.]

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01:48

Problem 51

Let $f:[a, b] \rightarrow \mathbb{R}$ be increasing and define $g:[a, b] \rightarrow \mathbb{R}$ by $g(x)=$ $f(x+)$ for $a \leq x<b$ and $g(b)=f(b)$. Prove that $g$ is increasing and rightcontinuous.

Gregory Higby

Numerade Educator