Math 104 Assignment 7

I recommend thinking about the first "Not definitions" problem below.

1. Limits and order properties

   Clarification: in this question and the rest of the course, unless otherwise specified, the metric we take on $\R$ is the usual distance: $d(x, y) = \abs{y-x}$.
   1. Suppose that $(a_n)_n$ is a convergent sequence in $\R$, and there are $N \in \N$ and $b \in \R$ such that $a_n \leq b$ for all $n \gt N$. Prove that \[\lim_{n\to\infty} a_n \leq b.\]
   2. Suppose that $(a_n)_n$ and $(b_n)_n$ are convergent sequences in $\R$ such that for some $N \in \N$ and every $n \gt N$, $a_n \leq b_n$. Prove that \[\lim_{n\to\infty} a_n \leq \lim_{n \to \infty} b_n.\] (Hint: use Part A.)
   3. Suppose that $(a_n)_n$ and $(b_n)_n$ are convergent sequences in $\R$ such that for some $N \in \N$ and every $n \gt N$, $a_n \leq b_n$, and so that \[\lim_{n\to\infty}a_n = \lim_{n\to\infty}b_n.\] Suppose further that $(c_n)_n$ is another sequence so that $a_n \leq c_n \leq b_n$ for $n \gt N$. Prove that $(c_n)_n$ converges to the same limit as $(a_n)_n$ and $(b_n)_n$.
   4. Show by example that if $(a_n)_n$ is a convergent sequence in $\R$ and $b \in \R$ is such that $a_n \lt b$ for all $n$, it may not be the case that \[\lim_{n\to\infty} a_n \lt b.\]
   5. Show that if $(a_n)_n$ is Cauchy in some metric space $(M, d)$, then \[\lim_{n\to\infty}\paren{\lim_{k\to\infty} d(a_n, a_k)} = 0.\]
   6. Show that if $(a_n)_n$ is a sequence in some metric space $(M, d)$, then $(a_n)_n$ converges to some $x \in M$ if and only if \[\lim_{n\to\infty} d(a_n, x) = 0.\]

2. Examining completions

   Let $(M, d)$ be a metric space. Recall that we constructed the completion of $M$ as \[\overline{M} = \set{\text{Cauchy sequences in } M} / \sim,\] where $(a_n)_n \sim (b_n)_n$ if and only if \[\lim_{n\to\infty} d(a_n, b_n) = 0.\] We then considered an isometry (i.e., distance-preserving function) $i : M \hookrightarrow \overline{M}$ given by \[i(x) = [(x)_n]_\sim.\]
   1. Show that for any $x \in M$, \[i(x) = \set{(a_n)_n \mid (a_n)_n \text{ is a sequence in } M \text{ with } \lim_{n\to\infty}a_n = x}.\]
   2. We saw that $\overline{M}$ is complete, but let's examine how limits there actually behave. Show that if $(a_n)_n$ is a Cauchy sequence in $M$, \[\lim_{n\to\infty} i(a_n) = [(a_n)_n]_\sim.\] (That is, Cauchy sequences in $M$ give sequences in $\overline{M}$ which converge to the equivalence class of the original sequence in $M$.)

3. A characterisation of density

   Suppose that $(M, d)$ is a metric space and $X \subseteq M$. Prove that the following are equivalent:
   1. $X$ is dense in $M$, i.e., $M = \overline{X}$ (the closure of $X$);
   2. for every $a \in M$ and every $\epsilon \gt 0$ there is some $x \in X \cap B_\epsilon(a)$; and
   3. for every $a \in M$ there is some sequence in $X$ converging to $a$.
   (Warning: this is not the same as saying that every $a \in M$ is a limit point of $X$; convince yourself that these are different.)

4. Limits of functions

   1. Suppose $(M, d)$ is a metric space, $E \subseteq M$, $f, g : E \to \R$, and $p$ is a limit point of $E$. Suppose further that for some $r \gt 0$, the set $f(E\cap B_r(p)) = \set{f(x) \mid x \in E\cap B_r(p)} \subseteq \R$ is bounded, and that \[\lim_{x\to p}g(x) = 0.\] Show that \[\lim_{x\to p} f(x)g(x) = 0.\] (Note: you should not assume that $f$ has a limit at $p$, unless you prove it from these hypotheses.)

   Suppose that $(M, d)$ and $(X, d_X)$ are metric spaces, $E \subseteq M$, $f : E \to \color{red}{X}$, and $p$ is a limit point of $E$.

   2. Show that if $f$ has a limit at $p$, then for every $\epsilon \gt 0$ there is some $r \gt 0$ so that whenever $x, y \in B_r(p) \cap E \color{red}{\setminus \set{p}}$, $d_X(f(x), f(y)) \lt \epsilon$.
   3. Suppose that for every $\epsilon \gt 0$ there is some $r \gt 0$ so that whenever $x, y \in B_r(p) \cap E$, $d_X(f(x), f(y)) \lt \epsilon$. Show that if $X$ is complete, then $f$ has a limit at $p$.
   4. Show that part (C) may fail if $X$ is not assumed to be complete.

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.

• Not definitions

  1. We say a sequence $(a_n)_n$ in a metric space $(M, d)$ is glomvergent if for every $\epsilon \gt 0$ there is $L \in M$ and $N \in \N$ so that for every $n \gt N$, $d(a_n, L) \lt \epsilon$.
     1. Are all sequences glomvergent?
     2. Are all convergent sequences glomvergent?
     3. Are all glomvergent sequences convergent?
     4. Does a glomvergent sequence exist?
  2. We say a sequence $(a_n)_n$ in a metric space $(M, d)$ is revergent if there is some $L \in M$ so that for every $N \in \N$ and every $n \gt N$, we have $d(a_n, L) \lt d(a_N, L)$.
     1. Are all sequences revergent?
     2. Are all convergent sequences revergent?
     3. Are all revergent sequences convergent?
     4. Does a revergent sequence exist?
  3. We say a sequence $(a_n)_n$ in a metric space $(M, d)$ is trivergent if there is some $L \in M$ and $N \in \N$ so that for every $n \gt N$, for every $\epsilon \gt 0$, we have $d(a_n, L) \lt \epsilon$.
     1. Are all sequences trivergent?
     2. Are all convergent sequences trivergent?
     3. Are all trivergent sequences convergent?
     4. Does a trivergent sequence exist?
  4. We say a sequence $(a_n)_n$ in a metric space $(M, d)$ is provergent if there is some $L \in M$ so that for every $\epsilon \gt 0$ and $N \in \N$, $n \gt N$ implies $d(a_n, L) \lt \epsilon$.
     1. Are all sequences provergent?
     2. Are all convergent sequences provergent?
     3. Are all provergent sequences convergent?
     4. Does a provergent sequence exist?
  5. We say a subset of a metric space is splopen if it contains all of its interior points.
     1. Are all subsets of metric spaces splopen?
     2. Are all open sets splopen?
     3. Are all splopen sets open?
     4. Does a splopen set exist?
  6. We say a subset of a metric space is fhtagn if it contains none of its limit points.
     1. Are all subsets of metric spaces fhtagn?
     2. Are all open sets fhtagn?
     3. Are all fhtagn sets open?
     4. Does a fhtagn set exist?
  7. We say a subset $U$ of a metric space $(M, d)$ is tropen if for every $x \in M$ and every $r \gt 0$, $B_r(x) \subseteq U$.
     1. Are all subsets of metric spaces tropen?
     2. Are all open sets tropen?
     3. Are all tropen sets open?
     4. Does a tropen set exist?
  8. We say a subset $U$ of a metric space $(M, d)$ is hopen if for every $x \in U$ and every $r \gt 0$, $B_r(x) \subseteq U$.
     1. Are all subsets of metric spaces hopen?
     2. Are all open sets hopen?
     3. Are all hopen sets open?
     4. Does a hopen set exist?
  9. We say a subset of a metric space is posed if it is not open.
     1. Are all subsets of metric spaces posed?
     2. Are all closed sets posed?
     3. Are all posed sets closed?
     4. Does a posed set exist?
  10. We say a subset of a metric space is clopen if it is both closed and open.
      1. Are all subsets of metric spaces clopen?
      2. Are all open sets clopen?
      3. Are all closed sets clopen?
      4. Does a clopen set exist?

• The empty metric space is silly

  Let $d : \emptyset\times\emptyset \to \R_{\geq 0}$ be a metric on $\emptyset$.
  1. Show that $\emptyset \subseteq \emptyset$ is not bounded.
  2. Show that $\emptyset$ is compact.
  3. Let $(M, d_M)$ be a metric space. Are all compact subsets of $M$ bounded?

• Pseudometrics

  A pseudometric on a set $M$ is a function $p : M\times M \to \R_{\geq0} such that:
  • for every $x \in M$, $p(x, x) = 0$;
  • for every $x, y \in M$, $p(x, y) = p(y, x)$; and
  • for every $x, y, z \in M$, $p(x, z) \leq p(x, y) + p(y, z)$.
  1. Prove that if $p$ is a pseudometric on $M$, then the relation $\sim$ defined by $x \sim y$ iff $p(x, y) = 0$ is an equivalence relation.
  2. Given $a, b \in M/\sim$ equivalence classes, let $x \in a$ and $y \in b$ and define $d(a, b) = p(x, y)$. Prove that $d$ does not depend on the choice of $x$ or $y$ here.
  3. Prove that $d$ is a metric on $M/\sim$.
  4. Let $p : \R^2\times\R^2 \to \R_{\geq 0}$ be defined by $p(x, y) = |x_1 + x_2 - y_1 - y_2|$. Show that $p$ is a pseudometric. Describe $\R^2/\sim$.
  5. Let $p : \R \times \R \to \R_{\geq 0}$ be defined by $p(x, y) = \inf\set{|x-y-k| \mid k \in \Z}$. Show that $p$ is a pseudometric. Describe $\R/\sim$.

• Families of pseudometrics

  1. Prove that in a metric space $(M, d)$, a sequence $(a_n)_n$ converges to a point $a$ if and only if for every open set $U \subseteq M$ with $a \in U$, there is some $N \in \N$ so that for every $n \gt N$, $a_n \in U$.

  Suppose that $M$ is a set, and let $\mathscr{F}$ be a set of pseudometrics on $M$. Let us call a subset $G \subseteq M$ open if for every $x \in M$, there are finitely many $p_1, \ldots, p_k \in \mathscr{F}$ and some $r \gt 0$ so that \[\set{y \in M \mid p_i(x, y) \lt r \text{ for each } i = 1, \ldots, k} \subseteq G.\]

  2. Prove that a set $G \subseteq M$ is open if and only if it is a union of finite intersections of sets of the form \[B_{p_i, r}(x) = \set{y \in M \mid p_i(x, y) \lt r}.\]

  We say a sequence $(a_n)_n$ in $M$ converges to $a \in M$ if for every open set $U \subseteq M$ with $a \in U$ there is some $N \in \N$ so that for every $n \gt N$, $a_n \in U$.

  3. Prove that every sequence in $M$ has at most one limit if and only if for every $x, y \in M$ there is some $p \in \mathscr{F}$ so that $p(x, y) \gt 0$.
  4. Let $M$ be the set of functions $\R \to \R$, and for each $x \in \R$, set \[p_x(f, g) = \abs{f(x) - g(x)}.\] Let $\mathscr{F} = \set{p_x \mid x \in \R}$. Show that a sequence of functions $(f_n)_n$ in $M$ converges to some $f \in M$ if and only if for each $x \in \R$, \[\lim_{n\to\infty} f_n(x) = f(x).\] (In this case, $(f_n)_n$ is said to converge pointwise to $f$.)