Math 104 Assignment 6

1. Distances and convergent sequences

   Let $(M, d)$ be a metric space.
   1. Let $(x_n)_n$ be a sequence in $M$, and $L \in M$. Prove that $(x_n)_n$ converges to $L$ if and only the sequence of real numbers $(d(x_n, L))_n$ converges to $0$.
   2. Suppose that $(a_k)_k$ and $(b_k)_k$ are sequences in $M$ which converge to $a$ and $b$ repsectively. Prove that \[\limni[{\color{red}k}] d(a_k, b_k) = d(a, b).\]

2. An interesting sequence of rational numbers

   Let $(q_n)_n$ be a sequence in $\Q_{\geq0}$ defined as follows: \[q_n = \begin{cases} \frac{a}{b} & \text{ if } n = 2^a3^b \text{ for some }a, b \in \N \\ 0 & \text{otherwise} \end{cases}.\] Show that for any $x \in \R_{\geq0}$ there is a subsequence of $(q_n)$ converging to $x$. (Hint: first show that every positive rational number occurs infinitely often in $(q_n)_n$, and recall that $\Q$ is dense in $\R$.)

3. Subsubsequences

   1. Suppose that $(M, d)$ is a metric space and $a \in M$. Suppose further that $(x_k)_k$ is a sequence in $M$ with the property that every subsequence of $(x_k)_k$ has a further subsequence which converges to $a$. Show that $(x_k)_k$ converges to $a$.
   2. Give an example of a sequence $(y_k)_k$ in some metric space, which does not converge but has the property that every subsequence has a further subsequence which does converge.

4. Square roots and convergent sequences

   1. Suppose that $(z_a)_a$ is a sequence in $\R_{\geq0}$ with \[\lim_{s\to\infty} z_s = z.\] Show that $(\sqrt{z_n})_n$ converges to $\sqrt z$. (Hint: it may be useful to treat the case $z = 0$ separately; it may also be useful to use the fact that $(\sqrt{z_t} - \sqrt{z})(\sqrt{z_t}+\sqrt{z}) = z_t - z$.)

   Let $a_1 = 1$ and for $n \in \N$, define $a_{n+1} = \sqrt{5+4a_n}$.

   2. Prove that for every $n \in \N$, we have $0 \leq a_n \leq a_{n+1} \leq 500!$.
   3. We will see later that Part A means $(a_n)_n$ must converge (monotonic sequences converge if and only if they are bounded). Assume that the limit of $(a_n)_n$ exists, and determine its value.

5. A compact set

   Suppose that $(a_n)_n$ is a sequence in a metric space $(M, d)$ which converges to a limit $a$. Prove that $\set{a_n\mid n \in \N} \cup \set a$ is compact.

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.

• Distances and convergent sequences

  Suppose that $(a_n)_n$, $(b_n)_n$ are sequences in a metric space $(M, d)$ so that \[\limni d(a_n, b_n) = 0.\] Prove that both sequences converge to the same limit, or provide a counterexample.

• An interesting sequence of rational numbers

  Let $(q_n)_n$ be the sequence from Question 2, and let $(a_n)_n$ be any sequence in $\Q$. Show that $(a_n)_n$ is a subsequence of $(q_n)_n$.

• Common subsequences

  Let $(M, d)$ be a compact metric space.
  1. Suppose that $(x_n)_n, (y_n)_n$ are sequences in $M$. Show that there is a common subsequence along which they both converge. (That is, show that there is an increasing sequence $(n_k)_k$ in $\N$ so that $(x_{n_k})_k$ and $(y_{n_k})_k$ both converge.)
  2. Suppose that $f_1, \ldots, f_T$ are sequences in $M$. (We might also write $(f_1(n))_n, (f_2(n))_n, \ldots, (f_T(n))_n$, but the double-indexing gets annoying.) Show that there is a common subsequence along which they all converge. (That is, show that there is an increasing function $n : \N \to \N$ so that $f_1\circ n, \ldots, f_T\circ n$ all converge; if we denote $n(k)$ by $n_k$, we might write $(f_1(n_k))_k, \ldots, (f_T(n_k))_k$ all converge.)
  3. Suppose that $(f_t)_t$ is a sequence of sequences in $M$ (so for each $t \in \N$, $f_t : \N \to M$). Show that there is a common subsequence along which they all converge. (That is, show that there is an increasing function $n : \N \to \N$ so that for all $t \in \N$, $f_t\circ n$ converges.)

• Topologies

  A topological space is a pair $(X, \mathscr{T})$ where $X$ is a set, and $\mathscr{T}$ is a collection of subsets of $X$ so that $\emptyset, X \in \mathscr{T}$, and $\mathscr{T}$ is closed under finite intersections and arbitrary unions (that is to say, if $G_1, \ldots, G_n \in \mathscr{T}$ then $G_1\cap\cdots\cap G_n \in \mathscr{T}$, and if $(G_\alpha)_\alpha$ is a collection of elements of $\mathscr{T}$ then $\bigcup_\alpha G_\alpha \in \mathscr{T}$). The elements of $\mathscr{T}$ are said to be open, and a set $F \subseteq X$ is said to be closed if $F^c \in \mathscr{T}$. We refer to $\mathscr{T}$ as a topology on $X$.

  Notice that if $(M, d)$ is a metric space, then $\mathscr{T}_d = \set{U \subseteq M \mid U \text{ is open with respect to } d}$ is a topology on $M$. We say that the topology $\mathscr{T}_d$ is induced by the metric $d$.

  A topological space $(X, \mathscr{T})$ is said to be metrizable if there is a metric on $X$ which induces $\mathscr{T}$. Note that this metric need not be unique!

  A sequence $(x_n)_n$ in a topological space $(X, \mathscr{T})$ is said to converge to $x \in X$ if for every open set $G \in \mathscr{T}$ with $x \in G$, there are only finitely many $n \in \N$ for which $x_n \notin G$. (Remember that we proved this was equivalent to our definition of convergence in metric spaces.)

  1. Let $X = \set{0, 1}$ and $\mathscr{T} = \set{\emptyset, X}$. Is $(X, \mathscr{T})$ a topological space? Is it metrizable?
  2. Are limits of sequences in topological spaces unique when they exist? Or can a sequence converge to more than one distinct point?
  3. Give an example of a topological space and two different metrics which give rise to the same topology.
  4. Formulate a necessary and sufficient condition for two metrics inducing the same topology.
  5. Suppose that $\mathscr{T}_1$ and $\mathscr{T}_2$ are topologies on a set $X$ with the property that any sequence in $X$ converges with respect to $\mathscr{T}_1$ if and only if it does with respect to $\mathscr{T}_2$. Must $\mathscr{T}_1 = \mathscr{T}_2$?