\[ \renewcommand{\C}{\mathbb{C}} \renewcommand{\R}{\mathbb{R}} \renewcommand{\Q}{\mathbb{Q}} \renewcommand{\Z}{\mathbb{Z}} \renewcommand{\N}{\mathbb{N}} \]
1 Least upper bounds
1.1 L02
Is the least upper bound of a subset of \(\Q\) or \(\R\) unique?
1.2 L02
Let \(x_{n}\in\Q\) be a non-decreasing sequence. Show that the least upper bound of the set \(\{x_n \colon n\in\N\}\) is also the limit of \(x_n\). In particular show that one exists if and only if the other one does.
2 Sequences and limits
2.1 L02
- In the definition of limit of a sequence, can we always assume \(\epsilon\lt 1\) instead of only \(\epsilon \gt 0\)?
- In the definition of limit of a sequence, can we always assume \(\epsilon=2^{-n}\) for some \(n\in\N\) instead of checking \(\forall \epsilon \gt 0\)?
- In the definition of limit of a sequence, can replace \(\forall \epsilon\gt 0 \ldots |x_{n}-L|\lt \epsilon\) with \(\forall \epsilon\gt 0 \ldots |x_{n}-L|\leq \epsilon\) ?
- In the definition of limit of a sequence, can replace \(\forall \epsilon\gt 0 \ldots |x_{n}-L|\lt \epsilon\) with \(\forall \epsilon\geq 0 \ldots |x_{n}-L|\leq \epsilon\) ?
2.2 L02
- Is a convergent sequence always bounded?
- Is a bounded sequence always convergent?
- harder Does a bounded sequence always have a limit point? In \(\R\)? In \(\Q\)?
2.3 L02
Using the words "eventually" and "frequently" negate the sentence "\(x_{n}\) eventually satisfies property \(P\)".
2.4 L02
Consider two sequences \(a_n\), \(b_n\), and the sequences \(a_n+b_n\), \(a_nb_n\), \(a_n/b_n\).
- Can you deduce the existence and value of the limits of the latter group from the existence and values of the limits of the former.
- Can you have the converse?
- What can you say about \(a_n\) if \(\frac{a_n}{b_n}\to L\in\R\) and \(b_n\to 0\)?
2.5 L03
We call the condition \(x \lt a\) an "open" condition on \(x\) and \(x\leq a\) a closed condition on \(x\).
- Is it true that if \(x_{n}\leq a\) eventually then the limit of \(x_n\), if it exists, also satisfies the closed condition (i.e. closed conditions are stable with respect to passing to the limit)?
- Is it true that if \(\lim_{n}x_{n}\lt a \) then \(x_{n}\lt a\) eventually (open conditions on the limit imply the open condition on the sequence)?
- Does the converse to the above statements hold?
2.6 L03
Is \(f(x)=x\) a continuous function from \(\Q\) to \(\R\)?
2.7 L01 L02
How do you prove the sandwich theorem for sequences?
2.7.1 L02
- Prove that \(\sum_{0}^{N}q^{n} = \frac{1-q^{N+1}}{1-q}\) (hint: Use induction)
- Prove that \(\sum_{0}^{\infty}q^{n} = \frac{1}{1-q}\) when \(|q|\lt 1\).
- Prove that \(\lim_{N\to \infty}\sum_{n=0}^{N} a_{n}q^{n}\) exists if \(|q| \lt 1\) and \(a_{n}\) is a bounded sequence.
3 Induction
3.1 L02
Use induction to show that \((1+x)^{n}\geq 1+nx\) for all \(|x|\leq 1\).
3.2 L02
Use induction to find a formula for \(\sum_{n=1}^{N} n^{2}\).
3.3 L02
Can you use induction to find a formula for \(\sum_{n=1}^{N} n^{d}\) for any \(d\in\N\)?
4 Constructing the Reals
4.1 L03
Recall the definition of a Cauchy sequence in \(\Q\).
4.2 L03
- Are all convergent sequences in \(\Q\) also Cauchy sequences?
- Are all Cauchy sequences in \(\Q\) convergent in \(\Q\)?
4.3 L03
Let \(\mathrm{Cauchy}(\Q)\) be the set of all Cauchy sequences in \(\Q\).
Let us introduce the relation
\begin{equation*} (x_n\in \Q)_{n\in\N} \sim_{\R} (y_n\in \Q)_{n\in\N} \text{ if } \lim_{n\in{\N}} |x_{n}-y_{n}|=0. \end{equation*}Is this an equivalence relation?
4.4 L03
We define \(\R=\mathrm{Cauchy}(\Q)/\sim_{\R}\)
- Using our definition of \(\R\), what is the meaning of a decimal expansion of a real?
- Why does any decimal expansion that we write according to the rules:
- use only digits \(0 \ldots 9\),
- after finitely many digits put a
decimal point
, - continue to infinity.
- According to our definition or \(\R\) explain why \(0.99999\ldots=1\).
4.5 L03
What is the role of Cauchy sequences with geometric control on differences in the construction of \(\R\) as a quotient space of Cauchy sequences valued in \(\Q\).
4.6 L04
Let \(\Q\subset X \subset \R\) and suppose that any bounded subset of \(X\) has a least upper bound in \(X\). Can you deduce that \(X=\R\)?