Sheet 1 Moodle version

This is the same content as the pdf in the Sheet 1 assignment, but visible as a Moodle page.

Q1

Prove, from the given axioms for the real numbers, that for real numbers \(a\), \(b\), \(c\), \(d\)

1. \(a(bc) = c(ba)\);
2. \(-(a+b) = (-a) + (-b)\);
3. if \(ab = ac\) and \(a \neq 0\), then \(b = c\);
4. if \(a < b\) and \(c < d\) then \(a + c < b + d\);
5. if \(a \leq b\) and \(c \leq d\) then \(a+c = b+d\) only if \(a = b\) and \(c = d\).

[Try to write out detailed answers, justifying each line of your argument by appealing to one of the axioms.]

Q2

Prove the following assertions, for real numbers \(a\), \(b\), \(c\):

1. if \(a < b\), then \(ac > bc\) if and only if \(c < 0\);
2. \(a^2 + b^2 = 0\) if and only if \(a = b = 0\);
3. \(a^3 < b^3\) if and only if \(a < b\).

[Less detailed answers are required than in Q1, but you should try to justify each step using axioms or results which have already been proved from the axioms.]

Q3

1. Prove that \((a^m)^{-1} = (a^{-1})^m\), for all \(a \in \mathbb{R}\setminus \{0\}\) (for \(m = 1\), \(2\), ...).
2. Prove that \(a^{k+1} = a^k a\) for \(a \neq 0\) and \(k = -1\), \(-2\), \(-3\), ....
3. Derive the law of indices: \(a^m a^n = a^{m+n}\) for \(a \neq 0\) and \(m\), \(n \in \mathbb{Z}\).

Q4

[In this question you may use familiar results about arithmetic and order in the real numbers. Here, and from now on, you are not expected to justify each line of your argument by citing axioms or properties derived from the axioms.]

For \(n = 1\), \(2\), \(3\), ..., let

\[a_n = \left(1 + \frac{1}{n}\right)^n \quad \textrm{ and } \quad b_n = \left(1 + \frac{1}{n}\right)^{n+1}. \]

1. Show that the inequality \(a_n \leq a_{n+1}\) can be rearranged as \[ \left(\frac{n(n+2)}{(n+1)^2}\right)^{n+1} \geq \frac{n}{n+1}. \] By applying Bernoulli's inequality to the left-hand side, verify this inequality.
2. Show that \(b_{n+1} \leq b_n\) for all \(n\).
3. Show that \(a_n \leq a_{n+1} \leq b_{n+1} \leq b_n\) for all \(n\). Deduce that \(a_n < 3\) for all \(n\).

[The significance of the result in this exercise will be revealed later!]

Q5

Define \(\max(a,b)\), the maximum of two real numbers \(a\) and \(b\), saying which axiom(s) show that your specification is well defined.  Using your definition, prove that \(\displaystyle \max(a,b) = \frac{1}{2}(a+b) + \frac{1}{2}|a-b|\), and write down an analogous formula for \(\min(a,b)\).