4.11.E: Problems on Double Limits and Product Spaces
Prove Theorem \(1(\mathrm{i}) .\) Prove Theorem 1\((\text { ii })\) for both choices of \(\rho,\) as suggested.
Formulate Definitions 2 and 3 for the cases
(i) \(p=q=s=+\infty\);
(ii) \(p=+\infty, q \in E^{1}, s=-\infty\);
(iii) \(p \in E^{1}, q=s=-\infty ;\) and
(iv) \(p=q=s=-\infty\).
Prove Theorem \(2^{\prime}\) from Theorem 2 using Theorem 1 of §2. Give a direct proof as well.
Define \(f : E^{2} \rightarrow E^{1}\) by
\[
f(x, y)=\frac{x y}{x^{2}+y^{2}} \text { if }(x, y) \neq(0,0), \text { and } f(0,0)=0 ;
\]
see §1, Example \((\mathrm{g}) .\) Show that
\[
\lim _{y \rightarrow 0} \lim _{x \rightarrow 0} f(x, y)=0=\lim _{x \rightarrow 0} \lim _{y \rightarrow 0} f(x, y) ,
\]
but
\[
\lim _{x \rightarrow 0 \atop y \rightarrow 0} f(x, y) \text { does not exist. }
\]
Explain the apparent failure of Theorem 2.
Define \(f : E^{2} \rightarrow E^{1}\) by
\[
f(x, y)=0 \text { if } x y=0 \text { and } f(x, y)=1 \text { otherwise. }
\]
Show that \(f\) satisfies Theorem 2 at \((p, q)=(0,0),\) but
\[
\lim _{(x, y) \rightarrow(p, q)} f(x, y)
\]
does not exist.
Do Problem \(4,\) with \(f\) defined as in Problems 9 and 10 of §3.
Define \(f\) as in Problem 11 of §3. Show that for \((\mathrm{c})\), we have
\[
\lim _{(x, y) \rightarrow(0,0)} f(x, y)=\lim _{x \rightarrow 0 \atop y \rightarrow 0} f(x, y)=\lim _{x \rightarrow 0} \lim _{y \rightarrow 0} f(x, y)=0 ,
\]
but \(\lim _{y \rightarrow 0} \lim _{x \rightarrow 0} f(x, y)\) does not exist; for \((\mathrm{d})\),
\[
\lim _{y \rightarrow 0} \lim _{x \rightarrow 0} f(x, y)=0 ,
\]
but the iterated limits do not exist; and for \((\mathrm{e}), \lim _{(x, y) \rightarrow(0,0)} f(x, y)\) fails to exist, but
\[
\lim _{x \rightarrow 0 \atop y \rightarrow 0} f(x, y)=\lim _{y \rightarrow 0} \lim _{x \rightarrow 0} f(x, y)=\lim _{x \rightarrow 0} \lim _{y \rightarrow 0} f(x, y)=0 .
\]
Give your comments.
Find (if possible) the ordinary, the double, and the iterated limits of \(f\) at \((0,0)\) assuming that \(f(x, y)\) is given by one of the expressions below, and \(f\) is defined at those points of \(E^{2}\) where the expression has sense.
\[
\begin{array}{ll}{\text { (i) } \frac{x^{2}}{x^{2}+y^{2}} ;} & {\text { (ii) } \frac{y \sin x y}{x^{2}+y^{2}}} \\ {\text { (iii) } \frac{x+2 y}{x-y} ;} & {\text { (iv) } \frac{x^{3} y}{x^{6}+y^{2}}} \\ {\text { (v) } \frac{x^{2}-y^{2}}{x^{2}+y^{2}} ;} & {\text { (vi) } \frac{x^{5}+y^{4}}{\left(x^{2}+y^{2}\right)^{2}}} \\ {\text { (vii) } \frac{y+x \cdot 2^{-y^{2}}}{4+x^{2}} ;} & {\text { (viii) } \frac{\sin x y}{\sin x \cdot \sin y}}\end{array}
\]
Solve Problem 7 with \(x\) and \(y\) tending to \(+\infty\).
Consider the sequence \(u_{m n}\) in \(E^{1}\) defined by
\[
u_{m n}=\frac{m+2 n}{m+n} .
\]
Show that
\[
\lim _{m \rightarrow \infty} \lim _{n \rightarrow \infty} u_{m n}=2 \text { and } \lim _{n \rightarrow \infty} \lim _{m \rightarrow \infty} u_{m n}=1 ,
\]
but the double limit fails to exist. What is wrong here? (See Theo\(\left.\text { rem } 2^{\prime} .\right)\)
Prove Theorem \(2,\) with (i) replaced by the weaker assumption ("subuni-form limit")
\[
(\forall \varepsilon>0)(\exists \delta>0)\left(\forall x \in G_{\neg p}(\delta)\right)\left(\forall y \in G_{\neg q}(\delta)\right) \quad \rho(g(x), f(x, y))<\varepsilon
\]
and with iterated limits defined by
\[
s=\lim _{x \rightarrow p} \lim _{y \rightarrow q} f(x, y)
\]
iff \((\forall \varepsilon>0)\)
\[
\left(\exists \delta^{\prime}>0\right)\left(\forall x \in G_{\neg p}\left(\delta^{\prime}\right)\right)\left(\exists \delta_{x}^{\prime \prime}>0\right)\left(\forall y \in G_{\neg q}\left(\delta_{x}^{\prime \prime}\right)\right) \quad \rho(f(x, y), s)<\varepsilon .
\]
Does the continuity of \(f\) on \(X \times Y\) imply the existence of (i) iterated limits? (ii) the double limit?
[Hint: See Problem 6.]
Show that the standard metric in \(E^{1}\) is equivalent to \(\rho^{\prime}\) of Problem 7 in Chapter 3, §11.
Define products of \(n\) spaces and prove Theorem 1 for such product spaces.
Show that the standard metric in \(E^{n}\) is equivalent to the product metric for \(E^{n}\) treated as a product of \(n\) spaces \(E^{1} .\) Solve a similar problem for \(C^{n} .\)
[Hint: Use Problem 13.]
Prove that \(\left\{\left(x_{m}, y_{m}\right)\right\}\) is a Cauchy sequence in \(X \times Y\) iff \(\left\{x_{m}\right\}\) and \(\left\{y_{m}\right\}\) are Cauchy. Deduce that \(X \times Y\) is complete iff \(X\) and \(Y\) are.
Prove that \(X \times Y\) is compact iff \(X\) and \(Y\) are.
[Hint: See the proof of Theorem 2 in Chapter 3, §16, for \(E^{2}\).]
(i) Prove the uniform continuity of projection maps \(P_{1}\) and \(P_{2}\) on \(X \times Y,\) given by \(P_{1}(x, y)=x\) and \(P_{2}(x, y)=y .\)
(ii) Show that for each open set \(G\) in \(X \times Y, P_{1}[G]\) is open in \(X\) and \(P_{2}[G]\) is open in \(Y .\)
[Hint: Use Corollary 1 of Chapter \(3,\{12 .]\)
(iii) Disprove (ii) for closed sets by a counterexample.
[Hint: Let \(X \times Y=E^{2} .\) Let \(G\) be the hyperbola \(x y=1 .\) Use Theorem 4 of Chapter 3, §16 to prove that \(G\) is closed.]
Prove that if \(X \times Y\) is connected, so are \(X\) and \(Y\).
[Hint: Use Theorem 3 of §10 and the projection maps \(P_{1}\) and \(P_{2}\) of Problem 17.]
Prove that if \(X\) and \(Y\) are connected, so is \(X \times Y\) under the product metric.
[Hint: Using suitable continuous maps and Theorem 3 in §10, show that any two "lines" \(x=p\) and \(y=q\) are connected sets in \(X \times Y .\) Then use Lemma 1 and Problem 10 in §10.]
Prove Theorem 2 under the weaker assumptions stated in footnote \(1 .\)
Prove the following:
(i) If
\[
g(x)=\lim _{y \rightarrow q} f(x, y) \text { and } H=\lim _{x \rightarrow p \atop y \rightarrow q} f(x, y)
\]
exist for \(x \in G_{\neg p}(r)\) and \(y \in G_{\neg q}(r),\) then
\[
\lim _{x \rightarrow p} \lim _{y \rightarrow q} f(x, y)=H .
\]
(ii) If the double limit and one iterated limit exist, they are necessarily equal.
In Theorem \(2,\) add the assumptions
\[
h(y)=f(p, y) \quad \text { for } y \in Y-\{q\}
\]
and
\[
g(x)=f(x, q) \quad \text { for } x \in X-\{p\} .
\]
Then show that
\[
\lim _{(x, y) \rightarrow(p, q)} f(x, y)
\]
exists and equals the double limits.
[Hint: Show that here (5) holds also for \(x=p\) and \(y \in G_{\neg q}(\delta)\) and for \(y=q\) and \( x \in G_{\neg p}(\delta) . ]\)
From Problem 22 prove that a function \(f :(X \times Y) \rightarrow T\) is continuous at \((p, q)\) if
\[
f(p, y)=\lim _{x \rightarrow p} f(x, y) \text { and } f(x, q)=\lim _{y \rightarrow q} f(x, y)
\]
for \((x, y)\) in some \(G_{(p, q)}(\delta),\) and at least one of these limits is uniform.