The Real Numbers: Sets, Sequences, and Functions

For $a \neq 0$ and $b \neq 0$, show that $(a b)^{-1} = a^{-1} b^{-1}$.

Verify the following:

For a non-empty set of real numbers $E$, show that $\inf E = \sup E$ if and only if $E$ consists of a single point.

Let $a$ and $b$ be real numbers.

Use the Completeness Axiom to show that every non-empty set of real numbers that is bounded below has an infimum and that

For real numbers $a$ and $b$, verify the following:

Use an induction argument to show that for each natural number $n$, the interval ($n, n+1$) fails to contain any natural number.

Use an induction argument to show that if $n > 1$ is a natural number, then $n-1$ also is a natural number. Then use another induction argument to show that if $m$ and $n$ are natural numbers with $n > m$, then $n-m$ is a natural number.

Show that for any real number $r$, there is exactly one integer in the interval $[r, r+1)$.

Show that any non-empty set of integers that is bounded above has a largest member.

Show that the irrational numbers are dense in $\mathbb {R}$.

Show that each real number is the supremum of a set of rational numbers and also the supremum of a set of irrational numbers.

Show that if $r > 0$, then, for each natural number $n,(1+r)^{n} \geq 1+n \cdot r$.

Use induction arguments to prove that for every natural number $n$,

Show that the set $\mathbb {Z}$ of integers is countable.

Show that a set $A$ is countable if and only if there is a one-to-one mapping of $A$ to $\mathbb {N}$.

Use an induction argument to complete the proof of part (i) of Corollary 4.

Corollary 4(i): For each natural number $n$, the Cartesian product $\mathbb {N} \times \cdots \times \mathbb {N}$ ($n$ times) is countably infinite. The proof given establishes this only for $n = 2$, using the one-to-one mapping $g: \mathbb {N} \times \mathbb {N} \rightarrow \mathbb {N}$ defined by $g(m, n) = (m+n)^{2}+n$.

Prove Corollary 6 in the case of a finite family of countable sets.

Corollary 6: The union of a countable collection of countable sets is countable.

Let both $f: A \rightarrow B$ and $g: B \rightarrow C$ be one-to-one and onto. Show that the composition $g \circ f: A \rightarrow B$ and the inverse $f^{-1}: B \rightarrow A$ are also one-to-one and onto.

Use an induction argument to establish the pigeonhole principle.

Show that $2^{\mathrm{N}}$, the collection of all sets of natural numbers, is uncountable.

Show that the Cartesian product of a finite collection of countable sets is countable. Use the preceding problem to show that $\mathbb {N}^{\mathbb {N}}$, the collection of all mappings of $\mathbb {N}$ into $\mathbb {N}$, is not countable.

Show that a non-degenerate interval of real numbers fails to be finite.

Show that any two non-degenerate intervals of real numbers are equipotent.

Is the set $\mathbb {R} \times \mathbb {R}$ equipotent to $\mathbb {R}$ ?

Is the set of rational numbers open or closed?

What are the sets of real numbers that are both open and closed?

Find two sets $A$ and $B$ such that $A \cap B = \emptyset$ and $\bar{A} \cap \bar{B} \neq \emptyset$.

A point $x$ is called an accumulation point of a set $E$ provided it is a point of closure of $E \sim \left\{ x\right\}$.

A point $x$ is called an isolated point of a set $E$ provided there is an $r > 0$ for which $(x-r, x+r) \cap E = \left\{ x\right\}$. Show that if a set $E$ consists of isolated points, then it is countable.

A point $x$ is called an interior point of a set $E$ if there is an $r > 0$ such that the open interval ($x-r, x+r$) is contained in $E$. The set of interior points of $E$ is called the interior of $E$ denoted by int $E$. Show that

Show that the Nested Set Theorem is false if $F_{1}$ is unbounded.

Show that the assertion of the Heine-Borel Theorem is equivalent to the Completeness Axiom for the real numbers. Show that the assertion of the Nested Set Theorem is equivalent to the Completeness Axiom for the real numbers.

Show that the collection of Borel sets is the smallest $\sigma$-algebra that contains the closed sets.

Show that the collection of Borel sets is the smallest $\sigma$-algebra that contains intervals of the form $[a, b)$, where $a < b$.

Show that each open set is an $F_{\sigma }$ set.

We call an extended real number a cluster point of a sequence $\left\{ a_{n}\right\}$ if a subsequence converges to this extended real number. Show that $\liminf \left\{ a_{n}\right\}$ is the smallest cluster point of $\left\{ a_{n}\right\}$ and $\limsup \left\{ a_{n}\right\}$ is the largest cluster point of $\left\{ a_{n}\right\}$.

Prove Proposition 19.

Proposition 19: Let $\left\{ a_{n}\right\}$ and $\left\{ b_{n}\right\}$ be sequences of real numbers.

1. $\limsup \left\{ a_{n}\right\} = \ell \in \mathbb {R}$ if and only if for each $\epsilon > 0$, there are infinitely many indices $n$ for which $a_{n} > \ell - \epsilon$ and only finitely many indices $n$ for which $a_{n} > \ell + \epsilon$.

2. $\limsup \left\{ a_{n}\right\} = \infty$ if and only if $\left\{ a_{n}\right\}$ is not bounded above.

3. $\limsup \left\{ a_{n}\right\} = -\liminf \left\{ -a_{n}\right\}$.

4. A sequence of real numbers $\left\{ a_{n}\right\}$ converges to an extended real number $a$ if and only if $\liminf \left\{ a_{n}\right\} = \limsup \left\{ a_{n}\right\} = a$.

5. If $a_{n} \leq b_{n}$ for all $n$, then $\limsup \left\{ a_{n}\right\} \leq \limsup \left\{ b_{n}\right\}$.

Show that a sequence $\left\{ a_{n}\right\}$ is convergent to an extended real number if and only if there is exactly one extended real number that is a cluster point of the sequence.

Show that $\liminf a_{n} \leq \limsup a_{n}$.

Prove that if, for all $n, a_{n} \geq 0$ and $b_{n} \geq 0$, then

provided the product on the right is not of the form $0 \cdot \infty$.

Show that every real sequence has a monotone subsequence. Use this to provide another proof of the Bolzano-Weierstrass Theorem.

Let $p$ be a natural number greater than 1 , and $x$ a real number, $0 \leq x \leq 1$. Show that there is a sequence $\left\{ a_{n}\right\}$ of integers with $0 \leq a_{n} < p$ for each $n$ such that

and that this sequence is unique except when $x$ is of the form $q / p^{n}, 0 < q < p^{n}$, in which case there are exactly two such sequences. Show that, conversely, if $\left\{ a_{n}\right\}$ is any sequence of integers with $0 \leq a_{n} < p$, the series

converges to a real number $x$ with $0 \leq x \leq 1$. If $p = 10$, this sequence is called the decimal expansion of $x$. For $p = 2$ it is called the binary expansion; and for $p = 3$, the ternary expansion.

Prove Proposition 20.

Proposition 20: Let $\left\{ a_{n}\right\}$ be a sequence of real numbers.

1. The series $\sum_{k=1}^{\infty } a_{k}$ is summable if and only if for each $\epsilon > 0$, there is an index $N$ for which $\left|\sum_{k=n}^{n+m} a_{k}\right| < \epsilon$ for $n \geq N$ and any natural number $m$.

2. If the series $\sum_{k=1}^{\infty } \left|a_{k}\right|$ is summable, then $\sum_{k=1}^{\infty } a_{k}$ also is summable.

3. If each term $a_{k}$ is non-negative, then the series $\sum_{k=1}^{\infty } a_{k}$ is summable if and only if the sequence of partial sums is bounded.

Show that the assertion of the Bolzano-Weierstrass Theorem is equivalent to the Completeness Axiom for the real numbers. Show that the assertion of the Monotone Convergence Theorem is equivalent to the Completeness Axiom for the real numbers.

Let $E$ be a closed set of real numbers and $f$ a real-valued function that is defined and continuous on $E$. Show that there is a function $g$ defined and continuous on all of $\mathbb {R}$ such that $f(x) = g(x)$ for each $x \in E$.

Define the real-valued function $f$ on $\mathbb {R}$ by setting

At what points is $f$ continuous?

Let $f$ and $g$ be continuous real-valued functions with a common domain $E$.

Show that a Lipschitz function is uniformly continuous but there are uniformly continuous functions that are not Lipschitz.

A continuous function $\varphi$ on $[a, b]$ is called piecewise linear provided there is a partition $a = x_{0} < x_{1} < \cdots < x_{n} = b$ of $[a, b]$ for which $\varphi$ is linear on each interval $\left[x_{i}, x_{i+1}\right]$. Let $f$ be a continuous function on $[a, b]$ and $\epsilon$ a positive number. Show that there is a piecewise linear function $\varphi$ on $[a, b]$ with $\left|f(x)-\varphi (x)\right| < \epsilon$ for all $x \in [a, b]$.

Show that a non-empty set $E$ of real numbers is closed and bounded if and only if every continuous real-valued function on $E$ takes a maximum value.

Show that a set $E$ of real numbers is closed and bounded if and only if every open cover of $E$ has a finite subcover.

Show that a non-empty set $E$ of real numbers is an interval if and only if every continuous real-valued function on $E$ has an interval as its image.

Show that a monotone function on an open interval is continuous if and only if its image is an interval.

Let $f$ be a real-valued function defined on $\mathbb {R}$. Show that the set of points at which $f$ is continuous is a $G_{\delta }$ set.

Let $\left\{ f_{n}\right\}$ be a sequence of continuous functions defined on $\mathbb {R}$. Show that the set of points $x$ at which the sequence $\left\{ f_{n}(x)\right\}$ converges to a real number is the intersection of a countable collection of $F_{\sigma }$ sets.

Let $f$ be a continuous real-valued function on $\mathbb {R}$. Show that the inverse image with respect to $f$ of an open set is open, of a closed set is closed, and of a Borel set is Borel.

A sequence $\left\{ f_{n}\right\}$ of real-valued functions defined on a set $E$ is said to converge uniformly on $E$ to a function $f$ if given $\epsilon > 0$, there is an $N$ such that for all $x \in E$ and all $n \geq N$, we have $\left|f_{n}(x)-f(x)\right| < \epsilon$. Let $\left\{ f_{n}\right\}$ be a sequence of continuous functions defined on a set $E$. Prove that if $\left\{ f_{n}\right\}$ converges uniformly to $f$ on $E$, then $f$ is continuous on $E$.