Math 104 Assignment 7

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

1. Limits and order properties

   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 \[\lim_{n\to\infty}\paren{\lim_{k\to\infty} d(a_n, a_k)} = 0.\]

2. Properties of limits superior and inferior

   Recall that the limit superior of a sequence of real numbers $(x_n)_n$ is defined to be \[ \limsup_{n\to\infty} x_n = \begin{cases} \infty & \text{ if } (x_n)_n \text{ is not bounded above}\\ -\infty & \text{ if } (x_n)_n \text{ is bounded above but } \paren{ \sup\set{x_k \mid k \gt n} }_n \text{ is not bounded below}\\ \displaystyle\lim_{n\to\infty} \sup\set{x_k \mid k \gt n} & \text{ otherwise}. \end{cases} \]

   1. Verify that if $(x_n)_n$ is bounded above, then the sequence $\paren{ \sup\set{x_k \mid k \gt n} }_n$ is monotonic.
   2. Prove that for every $y \in \R$ with $y \gt \limsup_{n\to\infty} x_n$, there are only finitely many $n \in \N$ for which $x_n \gt y$.
   3. Prove that for every $y \in \R$ with $y \lt \limsup_{n\to\infty} x_n$, there are infinitely many $n \in \N$ for which $x_n \gt y$.
   4. Prove that if $(x_n)_n$ and $(y_n)_n$ are sequences in $\R$, then \[\limsup_{n\to\infty} x_n + \limsup_{n\to\infty} y_n \geq \limsup_{n\to\infty} x_n + y_n,\] provided that all the limits superior involved are finite.
   5. Show by example that the above inequality is not always an equality.

   Be sure to consider the possibilities that $\limsup_{n\to\infty} x_n$ is $\pm\infty$ in B and C. Notice also that B and C together give another characterization of the limit superior: it is the unique element of $\R \cup \set{\infty, -\infty}$ with both properties.

3. 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, $E \subseteq M$, $f : E \to X$, 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 $x, y \in B_r(p) \cap E \setminus \set{p}$, $d_X(f(x), f(y)) \lt \epsilon$.
   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. They are coloured by approximate difficulty: easy/medium/hard.

• 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?

• More on limits superior and inferior

  1. Repeat problem 2 for limits inferior, adjusting inequalities as appropriate.
  2. Determine what can be said about 2.D. when some of the limits involved are not finite. Find all cases involving infinite limits where the inequality can still be established.
  3. Given $E$ a subset of a metric space $M$ with a limit point $p$ and $f : E \to \R$, propose a definition of \[\limsup_{x \to p} f(x),\] and prove that your definition has several nice properties.
  4. Suppose that $(x_n)_n$ is a sequence in $\R$. Show that $(x_n)_n$ converges if and only if \[\limsup_{n\to\infty} x_n = \liminf_{x\to\infty} x_n\] and both are finite.

• Rudin's definitions of limits superior and inferior

  The goal of this problem is to show that our definition of $\limsup$ and $\liminf$ agrees with Rudin's.

  Suppose that $(x_n)_n$ is a sequence in $\R$. We say that $(x_n)_n$ diverges to infinity and write \[\limni x_n = \infty\] if for every $M \in \R$ there is some $N \in \N$ so that for all $n \gt N$, $x_n \gt M$. Similarly, we say that $(x_n)_n$ diverges to negative infinity and write \[\limni x_n = -\infty\] if for every $M \in \R$ there is some $N \in \N$ so that for all $n \gt N$, $x_n \lt M$.

  Once again, suppose $(x_n)_n$ is a sequence in $\R$. Let $E$ be the set of all subsequential limits of $(x_n)_n$, including (potentially) $\infty$ and $-\infty$. Define \[\rls_{n\to\infty} x_n = \sup(E) \hspace{1in}\text{and}\hspace{1in} \rli_{n\to\infty} x_n = \inf(E),\] where the $\sup$ of any set containing $\infty$ is understood to be $\infty$, and the $\inf$ of any set containing $-\infty$ is likewise understood to be $-\infty$, and $\sup\set{-\infty} = -\infty$, $\inf\set\infty = \infty$.

  1. Let $(x_n)_n$ and $E$ be as above. Prove that if the set of (real) subsequential limits of $(x_n)_n$ is not bounded above, then $\infty \in E$. (To rephrase: if $E\cap \R$ is not bounded above, then $\infty \in E$.)
  2. Prove that $E$ is non-empty.
  3. Show that $\rls_{n\to\infty} x_n = \infty$ if and only if $\limsup_{n\to\infty} x_n = \infty$.
  4. Show that $\rls_{n\to\infty} x_n = L \in \R$ if and only if $\limsup_{n\to\infty} x_n = L$.
  5. Show that $\rls_{n\to\infty} x_n = -\infty$ if and only if $\limsup_{n\to\infty} x_n = -\infty$.

• Pseudometrics

  (The hardest part of this problem is understanding the definitions.)

  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 $x, y, z \in M$, $p(x, z) \leq p(x, y) + p(y, z)$.
  Note that we have not insisted that $p(x, y) = 0$ only if $y = x$.

  An equivalence relation on a set $X$ is a relation which is reflexive, symmetric, and transitive. That is, a relation $\sim$ such that:

  • for every $x \in X$, $x \sim x$;
  • for every $x, y \in X$, if $x \sim y$ then $y \sim x$; and
  • for every $x, y, z \in X$, if $x \sim y$ and $y \sim z$ then $x \sim z$.
  Given $x \in X$, the equivalence class of $x$ is the set $[x]_\sim = \set{y \in X \mid x \sim y}$. Notice that if $x \sim y$ then $[x]_\sim = [y]_\sim$. The set of equivalence classes is denoted \[{X/\!\sim} = \set{[x]_\sim \mid x \in X}.\]

  As an example, fix $k \in \N$ and define a relation $\sim$ on $\Z$ by $n \sim m$ if and only if $k$ divides $n - m$. Then, for example, $k \sim 15k \sim -2k$, and \[{\Z/\!\sim} = \set{[0]_\sim, [1]_\sim, \ldots, [k-1]_\sim}\] contains precisely $k$ distinct equivalence classes. It is possible to do arithmetic on these classes: we can define $[n]_\sim + [m]_\sim = [n+m]_\sim$ and $[n]_\sim[m]_\sim = [nm]_\sim$, and the definition does not depend on the choice of representatives $n \in [n]_\sim$, $m \in [m]_\sim$. This is the beginning of modular arithmetic.

  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

  Recall 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 \[\set{y \in M \mid p_i(x, y) \lt r \text{ for each } i = 1, \ldots, k} \subseteq G.\]

  1. 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 $n \gt N$, $a_n \in U$.

  2. 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 $p(x, y) \gt 0$.
  3. Let $\R^\R$ 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 $\R^\R$ converges to some $f \in \R^\R$ 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 $f$.)