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 oneone 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 oneone, 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 viceversa.
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.
