4.2.E: More Problems on Limits and Continuity
Complete the proof of Corollary 1.
\(\left[\text { Hint: Consider }\left\{f\left(x_{m}\right)\right\} \text { and }\left\{f\left(x_{m}^{\prime}\right)\right\}, \text { with }\right.\)
\[
x_{m} \rightarrow p \text { and } x_{m}^{\prime} \rightarrow p .
\]
By Chapter 3, §14, Corollary \(4, p\) is also the limit of
\[
x_{1}, x_{1}^{\prime}, x_{2}, x_{2}^{\prime}, \ldots ,
\]
so, by assumption,
\[
f\left(x_{1}\right), f\left(x_{1}^{\prime}\right), \ldots \text { converges (to } q, \text { say } ) .
\]
Hence \(\left\{f\left(x_{m}\right)\right\}\) and \(\left\{f\left(x_{m}^{\prime}\right)\right\}\) must have the same limit \(q .(\mathrm{Why} ?) ]\)
Complete the converse proof of Theorem 2 (cf. proof of Theorem 1 in
Chapter 3, §17).
Define \(f, g : E^{1} \rightarrow E^{1}\) by setting
(i) \(f(x)=2 ; g(y)=3\) if \(y \neq 2,\) and \(g(2)=0 ;\) or
(ii) \(f(x)=2\) if \(x\) is rational and \(f(x)=2 x\) otherwise; \(g\) as in \((\mathrm{i})\).
In both cases, show that
\[
\lim _{x \rightarrow 1} f(x)=2 \text { and } \lim _{y \rightarrow 2} g(y)=3 \text { but not } \lim _{x \rightarrow 1} g(f(x))=3 .
\]
Prove Theorem 3 from "\(\varepsilon, \delta\) " definitions. Also prove (both ways) that if \(f\) is relatively continuous on \(B,\) and \(g\) on \(f[B],\) then \(g \circ f\) is relatively continuous on \(B\).
Complete the missing details in Examples (A) and (B).
[Hint for (B): Verify that
\[
\left(1-\frac{1}{n+1}\right)^{-n-1}=\left(\frac{n}{n+1}\right)^{-n-1}=\left(\frac{n+1}{n}\right)^{n+1}=\left(1+\frac{1}{n}\right)\left(1+\frac{1}{n}\right)^{n} \rightarrow e . ]
\]
\(\Rightarrow 6 .\) Given \(f, g, h : A \rightarrow E^{*}, A \subseteq(S, \rho),\) with
\[
f(x) \leq h(x) \leq g(x)
\]
for \(x \in G_{\neg p}(\delta) \cap A\) for some \(\delta>0 .\) Prove that if
\[
\lim _{x \rightarrow p} f(x)=\lim _{x \rightarrow p} g(x)=q ,
\]
also then
\[
\lim _{x \rightarrow p} h(x)=q.
\]
Use Theorem 1.
[Hint: Take any
\[
\left\{x_{m}\right\} \subseteq A-\{p\} \text { with } x_{m} \rightarrow p .
\]
Then \(f\left(x_{m}\right) \rightarrow q, g\left(x_{m}\right) \rightarrow q,\) and
\[
\left(\forall x_{m} \in A \cap G_{\neg p}(\delta)\right) \quad f\left(x_{m}\right) \leq h\left(x_{m}\right) \leq g\left(x_{m}\right) .
\]
Now apply Corollary 3 of Chapter \(3, §15 . ]\)
\(\Rightarrow 7 .\) Given \(f, g : A \rightarrow E^{*}, A \subseteq(S, \rho),\) with \(f(x) \rightarrow q\) and \(g(x) \rightarrow r\) as \(x \rightarrow p\) \((p \in S),\) prove the following:
(i) If \(q>r,\) then
\[
(\exists \delta>0)\left(\forall x \in A \cap G_{\neg p}(\delta)\right) \quad f(x)>g(x) .
\]
(ii) (Passage to the limit in inequalities.) If
\[
(\forall \delta>0)\left(\exists x \in A \cap G_{\neg p}(\delta)\right) \quad f(x) \leq g(x) ,
\]
then \(q \leq r .\) (Observe that here \(A\) clusters at \(p\) necessarily, so the limits are unique.)
[Hint: Proceed as in Problem \(6 ;\) use Corollary 1 of Chapter \(3, §15 .]\)
Do Problems 6 and 7 using only Definition 2 of §1.
\([\text { Hint: Here prove } 7(\text { ii }) \text { first. }]\)
Do Examples \((a)-(d)\) of §1 using Theorem 1.
[Hint: For \((\mathrm{c}),\) use also Example (a) in Chapter \(3, §16 . ]\)
Addition and multiplication in \(E^{1}\) may be treated as functions
\[
f, g : E^{2} \rightarrow E^{1}
\]
with
\[
f(x, y)=x+y \text { and } g(x, y)=x y .
\]
Show that \(f\) and \(g\) are continuous on \(E^{2}\) (see footnote 2 in Chapter 3 §15). Similarly, show that the standard metric
\[
\rho(x, y)=|x-y|
\]
is a continuous mapping from \(E^{2}\) to \(E^{1}\).
\([\text { Hint: Use Theorems } 1,2, \text { and, } 4 \text { of Chapter } 3, §15 \text { and the sequential criterion. }]\)
Using Corollary 2 and formula \((9),\) find \(\lim _{x \rightarrow 0}(1 \pm m x)^{1 / x}\) for a fixed \(m \in N .\)
\(\Rightarrow 12 .\) Let \(a>0\) in \(E^{1} .\) Prove that \(\lim _{x \rightarrow 0} a^{x}=1\).
\(\left[\text { Hint: Let } n=f(x) \text { be the integral part of } \frac{1}{x}(x \neq 0) . \text { Verify that }\right.\)
\[
a^{-1 /(n+1)} \leq a^{x} \leq a^{1 / n} \text { if } a \geq 1 ,
\]
with inequalities reversed if \(0<a<1 .\) Then proceed as in Example \((\mathrm{A}),\) noting that
\[
\lim _{n \rightarrow \infty} a^{1 / n}=1=\lim _{n \rightarrow \infty} a^{-1 /(n+1)}
\]
by Problem 20 of Chapter \(3, §15 .(\text { Explain! })]\)
\(\Rightarrow 13 .\) Given \(f, g : A \rightarrow E^{*}, A \subseteq(S, \rho),\) with
\[
f \leq g \quad \text { for } x \text { in } G_{\neg p}(\delta) \cap A .
\]
Prove that
(a) if \(\lim _{x \rightarrow p} f(x)=+\infty,\) then also \(\lim _{x \rightarrow p} g(x)=+\infty\);
(b) if \(\lim _{x \rightarrow p} g(x)=-\infty,\) then also \(\lim _{x \rightarrow p} f(x)=-\infty\).
Do it it two ways:
(i) Use definitions only, such as \(\left(2^{\prime}\right)\) in \(§1\).
(ii) Use Problem 10 of Chapter 2, §13 and the sequential criterion.
\(\Rightarrow 14 .\) Prove that
(i) if \(a>1\) in \(E^{1},\) then
\[
\lim _{x \rightarrow+\infty} \frac{a^{x}}{x}=+\infty \text { and } \lim _{x \rightarrow+\infty} \frac{a^{-x}}{x}=0 ;
\]
(ii) if \(0<a<1,\) then
\[
\lim _{x \rightarrow+\infty} \frac{a^{x}}{x}=0 \text { and } \lim _{x \rightarrow+\infty} \frac{a^{-x}}{x}=+\infty ;
\]
(iii) if \(a>1\) and \(0 \leq q \in E^{1},\) then
\[
\lim _{x \rightarrow+\infty} \frac{a^{x}}{x^{q}}=+\infty \text { and } \lim _{x \rightarrow+\infty} \frac{a^{-x}}{x^{q}}=0 ;
\]
\((\text { iv })\) if \(0<a<1\) and \(0 \leq q \in E^{1},\) then
\[
\lim _{x \rightarrow+\infty} \frac{a^{x}}{x^{q}}=0 \text { and } \lim _{x \rightarrow+\infty} \frac{a^{-x}}{x^{q}}=+\infty .
\]
[Hint: (i) From Problems 17 and 10 of Chapter \(3, §15,\) obtain
\[
\lim \frac{a^{n}}{n}=+\infty .
\]
Then proceed as in Examples \((\mathrm{A})-(\mathrm{C}) ;\) (iii) reduces to (i) by the method used in
Problem 18 of Chapter \(3, §15 . ]\)
\(\Rightarrow * 15 .\) For a map \(f :(S, \rho) \rightarrow\left(T, \rho^{\prime}\right),\) show that the following statements are equivalent:
(i) \(f\) is continuous on \(S\).
(ii) \((\forall A \subseteq S) f[\overline{A}] \subseteq \overline{f[A]}\).
(iii) \((\forall B \subseteq T) f^{-1}[\overline{B}] \supseteq \overline{f^{-1}[B]}\).
(iv) \(f^{-1}[B]\) is closed in \((S, \rho)\) whenever \(B\) is closed in \(\left(T, \rho^{\prime}\right)\).
(v) \(f^{-1}[B]\) is open in \((S, \rho)\) whenever \(B\) is open in \(\left(T, \rho^{\prime}\right)\).
[Hints: \( (i) \Longrightarrow(\mathrm{ii})\): Use Theorem 3 of Chapter } 3, §16 and the sequential criterion to show that
\[
p \in \overline{A} \Longrightarrow f(p) \in \overline{f[A]} .
\]
(ii) \(\Longrightarrow(\text { iii }) :\) Let \(A=f^{-1}[B] .\) Then \(f[A] \subseteq B,\) so by \((\text { ii })\),
\[
f[\overline{A}] \subseteq \overline{f[A]} \subseteq \overline{B} .
\]
Hence
\[
\overline{f^{-1}[B]}=\overline{A} \subseteq f^{-1}[f[\overline{A}]] \subseteq f^{-1}[\overline{B}] . \quad \text { (Why?) }
\]
(iii) \(\Longrightarrow(\mathrm{iv}) :\) If \(B\) is closed, \(B=\overline{B}\) (Chapter 3, §16, Theorem 4(ii)), so by (iii),
\[
f^{-1}[B]=f^{-1}[\overline{B}] \supseteq \overline{f^{-1}[B]} ; \text { deduce (iv) } .
\]
\((\mathrm{iv}) \Longrightarrow(\mathrm{v}) :\) Pass to complements in \((\mathrm{iv})\).
\((\mathrm{v}) \Longrightarrow(\mathrm{i}) :\) Assume \((\mathrm{v}) .\) Take any \(p \in S\) and use Definition 1 in \(§1 . ]\)
Let \(f : E^{1} \rightarrow E^{1}\) be continuous. Define \(g : E^{1} \rightarrow E^{2}\) by
\[
g(x)=(x, f(x)) .
\]
Prove that
(a) \(g\) and \(g^{-1}\) are one to one and continuous;
(b) the range of \(g,\) i.e., the set
\[
D_{g}^{\prime}=\left\{(x, f(x)) | x \in E^{1}\right\} ,
\]
is closed in \(E^{2}\).
[Hint: Use Theorem 2 of Chapter 3, §15, Theorem 4 of Chapter 3, §16, and the sequential criterion.]