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Real Analysis: Math 4050-4060

Valerio De Angelis

Section 2.3 Exercises

Exercise 2.3.1.

Define the operations of addition and multiplication on the set \({\mathbb Q}\times {\mathbb Q}\) by
\begin{equation*} (a_1,b_1)+(a_2,b_2)=(a_1+a_2,b_1+b_2), \end{equation*}
\begin{equation*} (a_1,b_1)(a_2,b_2)=(a_1a_2-b_1b_2,a_1b_2+b_1a_2). \end{equation*}
  1. Prove that the resulting set with these two operations is a field, with additive identity \((0,0)\) and multiplicative identity \((1,0)\text{.}\)
  2. Define \(i=(0,1)\text{.}\) Prove that \((a,0)+i(b,0) =(a,b)\text{,}\) and \(i^2=(-1,0)\text{.}\)
The set \({\mathbb Q}\times {\mathbb Q}\) together with the two operations defined above is denoted by \({\mathbb Q}[i]\text{,}\) and by writing \(x\) instead of \((x,0)\text{,}\) we obtain the familiar complex numbers \(a+ib\text{,}\) where \(i^2=-1\) and \(a,b\) are rational numbers.

Exercise 2.3.2.

Use only the field axioms to prove that the two solutions \(\frac{-b}{a}\) and \(-\frac{b}{a}\) of Example 2.1.1 are the same.

Exercise 2.3.3.

Find examples to show that subtraction and division are neither commutative nor associative.

Exercise 2.3.4.

Prove that the uniqueness of \(1, -x, x^{-1}\) follows from the other axioms for ordered fields.
Proof Suppose \(1'\) is another multiplicative identity. Then \(1=11'=1'\text{.}\) If \(y\) is an element of \(F\) such that \(x+y=0\text{,}\) then \((-x)+(x+y)=(-x)+0\text{,}\) \(((-x)+x)+y=-x\text{,}\) \(0+y=-x\text{,}\) \(y=-x\text{.}\) Similarly for \(x^{-1}\text{.}\)

Exercise 2.3.5.

Let \(F\) be an ordered field. Give an axiomatic proof of the following statements. Make sure you quote the relevant axioms for each step. Do the proofs of a.-d. in that order, because part b. may require part a., part c. may require part b., etc.
  1. \(-(-x)=x\) for all \(x\in F\text{.}\)
  2. \(\displaystyle (-1)^2 = 1\)
  3. If \(x\in F\) and \(x\neq 0\text{,}\) then \(x^2\gt 0\text{.}\)
  4. \(\displaystyle 0\lt 1\)

Exercise 2.3.6.

Let \(x,y,z\) be real numbers. Give an axiomatic proof of the following statements.
  1. If \(xy=xz\) and \(x\neq 0\text{,}\) then \(y=z\text{.}\)
  2. If \(x\gt 1\text{,}\) then \(x^2\gt x\text{.}\)
  3. If \(0\lt x\lt 1\text{,}\) then \(x^2\lt 1\text{.}\)
  4. If \(x\gt 0\text{,}\) then \(x^{-1}\gt 0\)

Exercise 2.3.7.

Prove that \(||x|-|y||\leq |x\pm y|\) for all real numbers \(x,y\text{.}\)

Exercise 2.3.8.

Let \(x_1, x_2, \ldots, x_n\) be real numbers. Prove by induction that
\begin{equation*} \left|\sum_{k=1}^n x_k\right| \leq \sum_{k=1}^n|x_k|. \end{equation*}

Exercise 2.3.9.

Let \(x,y\) be real numbers, and suppose that \(|x-y|\lt \varepsilon \) for all \(\varepsilon \gt 0\text{.}\) Prove that \(x=y\text{.}\)

Exercise 2.3.10.

Prove the following set equalities:
\begin{equation*} \bigcap_{n=1}^\infty \left(-\frac{2}{n}, 1+\frac{2}{n}\right)=[0,1] \end{equation*}
\begin{equation*} \bigcup_{n=1}^\infty \left[\frac{1}{n}, 1-\frac{1}{n}\right]=(0,1) \end{equation*}

Exercise 2.3.11.

Prove that if a subset \(S\) has a maximum \(M\text{,}\) then \(M\) is unique.

Exercise 2.3.12.

Suppose that \(S\) is a bounded subset of \({\mathbb R}\text{.}\) Prove that \(\sup S \in S\) if and only if the maximum of \(S\) exists, and \(\inf S \in S\) if and only if the minimum of \(S\) exists.

Exercise 2.3.13.

Consider the set
\begin{equation*} S=\left\{\frac{n}{n+1}: n\in {\mathbb N}\right\}. \end{equation*}
  1. Decide if the set is bounded above, bounded below, or neither.
  2. Decide is the set has a maximum, a minimum, or neither, and find maximum and minimum if they exist.
  3. If the set is bounded above, find its supremum.
  4. If the set is bounded below, find its infimum.

Exercise 2.3.14.

Repeat Exercise 2.3.13 for the set
\begin{equation*} S=\left\{1+\frac{(-1)^n}{n}: n\in {\mathbb N}\right\}. \end{equation*}

Exercise 2.3.15.

Suppose that \(S\) and \(T\) are subsets of \(\mathbb R\text{,}\) and \(T\subseteq S\text{.}\) Prove that \(\inf S \leq \inf T \leq \sup T \leq \sup S\text{.}\)

Exercise 2.3.16.

Let \(A\) and \(B\) be subsets of \(\mathbb R\text{,}\) and define \(A+B = \{a+b: a\in A, b\in B\}\text{.}\) Suppose that both \(A\) and \(B\) are bounded above. Prove that \(A+B\) is bounded above, and
\begin{equation*} \sup(A+B) = \sup A + \sup B. \end{equation*}

Exercise 2.3.17.

Let \(S=\{x\in {\mathbb Q}| x^2 \lt 2\}\text{.}\)
  1. Prove that \(2\) is an upper bound for \(S\text{.}\)
  2. Prove that if \(p^2\gt 2\text{,}\) then \(p\) is an upper bound for \(S\text{.}\)

Exercise 2.3.18.

Prove that if \(\alpha = \sup\{x\in {\mathbb R}: x^2\lt 2\}\text{,}\) then \(\alpha^2=2\text{.}\)

Exercise 2.3.19.

Prove that in \(\mathbb R\text{,}\) \(x^2\lt 2\) is equivalent to \(-\sqrt{2}\lt x \lt \sqrt{2}\text{.}\)

Exercise 2.3.20.

Show that if we assume that \(-\infty\) is the additive inverse of \(\infty\text{,}\) we can conclude that every real number is equal to \(0\text{.}\)

Exercise 2.3.21.

Suppose we want to define \(\sup \emptyset\text{,}\) and we can choose any element of \({\mathbb R}\cup \{\infty, -\infty\}\text{.}\) Show that the only choice that does not lead to a contradiction is \(-\infty\text{.}\)
Think of what \({\cal U}(\emptyset)\) must be.

Exercise 2.3.22.

Prove that the irrational numbers \({\mathbb R}\setminus {\mathbb Q}\) are dense.
Consider the set \(\{\sqrt{2}/n: n\in {\mathbb N}\}\text{.}\)
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