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 \(X=\R\)?