maa 4211 Fall19 homework

MAA4102 Homework


  • Homework 10. Suppose \(K\) is a nonempty compact subset of a metric space \(X\) and \(x\in X\).
    • Show, there is a nearest point \(p\in K\) to \(x;\) that is, there is a point \(p\in K\) such that, for all other \(q\in K\),
      \[
      d(p,x)\le d(q,x).
      \]
      [Suggestion: As a start, let \(S=\{d(x,y):y\in K\}\) and show there is a sequence \((q_n)\) from \(K\) such that the numerical sequence \((d(x,q_n))\) converges to \(\inf(S)\).]

    • Let \(X=\mathbb R^2\) and \(\mathbb T =\{(x,y):x^2+y^2=1\}.\) Show, there is
      a point \(z\in X\) and distinct points \(a,b\in \mathbb T\) that are nearest points to \(z.\)

    • Let \(X=(0,\infty) \cup \{-1\}\) (as a subspace of \(\mathbb R\)). Show there is no nearest point to \(-1\) from the closed (in \(X\)) set \(K=(0,1].\)

      [Latex code] Suppose $K$ is a nonempty compact subset of a metric space $X$ and $x\in X$.

      Show, there is a {\it nearest point} $p\in K$ to $x;$ that is,
      there is a point $p\in K$ such that, for all other $q\in K$, we have $d(p,x)\le d(q,x).$
      [Suggestion: As a start, let $S=\{d(x,y):y\in K\}$ and show there is
      a sequence $(q_n)$ from $K$ such that the numerical sequence $(d(x,q_n))$ converges to $\inf(S)$.]

      Let $X=\mathbb R^2$ and $\mathbb T =\{(x,y):x^2+y^2=1\}.$ Verify there is
      a point $z\in X$ and distinct points $a,b\in \mathbb T$ that are nearest points to $z.$

      Let $X=(0,\infty) \cup \{-1\}$ (as a subspace of $\mathbb R$). Verify there is
      no nearest point to $-1$ from the closed (in $X$) set $K=(0,1].$
      \end{enumerate}

  • Homework 9. Prove: A finite union of compact sets is compact. (Problem 6.2.) (30 Oct)
      Suppose $X$ is a metric space, $n\in\mathbb N^+$ and $K_1,\dots,K_n\subset X.$ Show, if each $K_j$ is compact, then $K=\cup_{j=1}^n K_j$ is compact.



  • Homework 8. Problem 4.6. [Suggestion: Show \( \widetilde{S^\prime} \) is open using Problem 4.5. You don’t need to include a proof of the first part of Problem 4.5.]
      Let $S^\prime$ denote the set of limit points of a subset $S$ of a metric space $X$. Prove $S^\prime$ is closed.

  • Homework 7. Problem 4.1. [Suggestion: First show, given \( N\in\mathbb N.\) the set If \(\sigma^{-1}(\{0,1,\dots,N\})\) is either empty or has a largest element \( M\) and if \( m> M\), then \( \sigma(m)> N,\) being careful to point out where the assumption that \( \sigma\) is one-one is used. Now conclude, given \( N\in \mathbb N\), there is an \( M\in \mathbb N\) such that if \( m > M,\) then \(\sigma(m) > N.\)]
      Suppose $(a_n)$, a sequence in a metric space $X$, converges to $L\in X$. Show, if $\sigma:\mathbb N\to\mathbb N$ is one-one, then the sequence $(b_n=a_{\sigma(n)})_n$ also converges to $L$.

  • Homework 6. Problem 3.9 plus determine the boundary of \( \mathbb Q \subset \mathbb R. \)
      Prove that $x\in\partial S$ if and only if for every $\epsilon>0$ there exists $s\in S$, $t\in \widetilde{S}$ such that $d(x,s),d(x,t)<\epsilon$. Prove $S$ is closed if and only if $S$ contains its boundary; and $S$ is open if and only if $S$ is disjoint from its boundary. Determine the boundary of $\mathbb Q.$

  • Homework 5. Given a metric space \( Z \) and \( F\subseteq X\subseteq Z\) define \( F \) is relatively closed in \(X.\) Show, \( F \) is relatively closed in \( X \) if and only if there is a closed set \( C\subseteq Z \) such that \( F=C\cap X.\) (25 Sep)

      Given a metric space $Z$ and $F\subseteq X\subseteq Z$ define $F$ is \emph{relatively closed} in $X$. Show, $F$ is relatively closed in $X$ if and only if there is a closed set $C\subseteq Z$ such that $F=C\cap X$.


  • Homework 4. Problem 3.1. (9 Sep)
      Suppose $(X,d_X)$ and $(Y,d_Y)$ are metric spaces. Define
      $d:(X\times Y)\times (X\times Y)\to \mathbb R$ by
      $$
      d((x,y),(a,b))=d_X(x,a)+d_Y(y,b).
      $$
      Prove $d$ is a metric on $X\times Y$.
  • Homework 3. Exercise 2.4 and Problem 2.3. (30 Aug)
      Prove that if $A\subseteq B$ are subsets of $\mathbb R$ and $A$ is nonempty and $B$ is bounded above, then $A$ and $B$ have least upper bounds and
      \begin{equation*}
      \sup(A)\le \sup(B).
      \end{equation*}

      Suppose $A\subseteq \mathbb R$ is nonempty and bounded above and $\beta\in \mathbb R$. Let
      \begin{equation*}
      A+\beta =\{a+\beta: a\in A\}
      \end{equation*}
      Prove that $A+\beta$ has a supremum and
      \begin{equation*}
      \sup(A+\beta) =\sup(A)+\beta.
      \end{equation*}

  • Homework 2. Problem 1.2. Due Friday 30 August. (26 Aug)
      Suppose $f:X\to S$ and $\mathcal F\subseteq P(S)$. Show,
      \[
      \begin{split}
      f^{-1}(\cup_{A\in \mathcal F} A)=&\cup_{A\in\mathcal F} f^{-1}(A) \\
      f^{-1}(\cap_{A\in \mathcal F} A)=&\cap_{A\in\mathcal F} f^{-1}(A)
      \end{split}
      \]

      Show, if $A,B\subseteq X$, then $f(A\cap B)\subseteq f(A)\cap f(B)$.
      Give an example, if possible, where strict inclusion holds.

      Show, if $C\subseteq X$, then $f^{-1}(f(C)) \supseteq C$. Give an
      example, if possible, where strict inclusion holds.

  • Homework 1. Problem 1.1. Due Monday 26 August. (21 Aug)



    Homework grading abbreviations

  • QU. Quantifier unclear.
  • QE. Quantifier error.
  • QI. The symbol needs an introduction with a quantifier.
  • IT. If needs a then, or vice-versa.
  • IDF. I don’t follow.
  • Circular. Assumes what is to be proved.
  • ETR. Easy to read. Thanks!
  • MC. Missing conjunction(s).
  • SF. Sentence fragment or not a sentence.
  • DNP. Parse error (doesn’t parse).
  • DNF. Does not follow.
  • NN. Narration needed.
  • NRD. Not relevant, delete.
  • ꟼ start new paragraph.