Exercise 7.17

The extensions of the definitions, and the statements and proofs of Theorems 7.8 through 7.11, are trivial, so I will simply copy them from the text, italicizing the changes.

Definition 7.7 We say that a sequence of mappings $\{f_n\}$, $n=1,2,3,\dots ,$into a metric space $F$ converges uniformly on $E$ to a function $f$ if for every $𝜀>0$ there is an integer $N$ such that $n\ge N$ implies

for all $x\in E$.

Definition 7.22 A family $F$ of mappings $f$ defined on a set $E$ in a metric space $X$with values in a metric space $Y$ is said to be equicontinuous on $E$ if for every $𝜀>0$ there exists a $\delta >0$ such that

whenever $d_X(x,y)\le \delta$, $x\in E$, $y\in E$, and $f\in F$.

Theorem 7.8 The sequence of mappings $\{f_n\}$into a complete metric space $F$, defined on $E$, converges uniformly on $E$ if and only if for every $𝜀>0$ there exists an integer $N$ such that $m\ge N$, $n\ge N$, $x\in E$ implies $d\left(f_n(x),f_m(x)\right)\le 𝜀$.

Proof: Suppose $\{f_n\}$ converges uniformly on $E$, and let $f$ be the limit function. There there is an integer $N$ such that $n\ge E$ implies $d\left(f_n(x),f(x)\right)\le 𝜀∕2$ so that

if $n\ge N$, $m\ge M$, $x\in E$.

Conversely, suppose the Cauchy condition holds. Since $F$is a complete metric space, the sequence $\{f_n(x)\}$ converges, for every $x$, to a limit which we may call $f(x)$. Thus the sequence $\{f_n\}$ converges on $E$, to $f$. We have to prove that the convergence is uniform.

Let $𝜀>0$ be given, and choose $N$ such that (13) holds. Fix $n$, and let $m\to \infty$ in (13). Since $f_m(x)\to f(x)$ as $m\to \infty$, this gives $d\left(f_n(x),f(x)\right)\le 𝜀$ for every $n\ge N$ and every $x\in E$, which completes the proof.

Theorem 7.9 Suppose $\{f_n\}$is a sequence of mappings from $E$into a metric space $F$such that $\lim <!--\; FUNCTION\; APPLICATION\; -->f_n(x)=f(x)$ for $x\in E$. Put

Then $f_n\to f$ uniformly on $E$ if and only if $M_n\to 0$ as $n\to \infty$.

Proof: Since this is an immediate consequence of Definition 7.7, we omit the details of the proof.

Theorem 7.10 Suppose $\{f_n\}$ is a sequence of vector-valued functions with values in $R^k$ defined on $E$, and suppose

Then $\sum <!--\; FUNCTION\; APPLICATION\; -->f_n$ converges uniformly on $E$ if $\sum <!--\; FUNCTION\; APPLICATION\; -->M_n$ converges.

Proof: If $\sum <!--\; FUNCTION\; APPLICATION\; -->M_n$ converges, then, for arbitrary $𝜀>0$,

provided $m$ and $n$ are large enough. Uniform convergence now follows from Theorem 7.8.

Theorem 7.11 Suppose $\{f_n\}$, a sequence of mappings with values in a complete metric space $F$, converges uniformly on a set $E$ in a metric space. Let $x$ be a limit point of $E$, and suppose that

Then $\{A_n\}$ converges and

Proof: Let $𝜀>0$ be given. By the uniform convergence of $\{f_n\}$, there exists $N$ such that $n\ge N$, $m\ge N$, $t\in E$ imply

Letting $t\to x$ in (18), we obtain

for $n\ge N$, $m\ge M$, so that $\{A_n\}$ is a Cauchy sequence and therefore converges, say to $A$.

Next,

We first choose $n$ such that

for all $t\in E$ (this is possible by the uniform convergence), and such that

Then, for this $n$ we choose a neighborhood $V$ of $x$ such that

if $t\in V\cap E$, $t\ne x$.

Substituting the inequalities (20) to (22) into (19), we see that $d\left(f(t),A\right)\le 𝜀$, provided $t\in V\cap E$, $t\ne x$. This is equivalent to (16).

Theorem 7.12 If $\{f_n\}$ is a sequence of continuous mappings on $E$into a metric space $F$, and if $f_n\to f$ uniformly on $E$, then $f$ is continuous on $E$.

Proof: (In the text, Theorem 7.12 is an immediately corollary of Theorem 7.11. Here, that would require that $F$ be complete, which is not necessarily the case. So the following proof is entirely new.)

By Theorem 4.8, to show that $f$is continuous, it suffices to show that $f^{-1}(V)$ is open for any open set $V\subset F$. If $x\in f^{-1}(V)$, we need to find a neighborhood $N_x$ of $x$ which is contained in $f^{-1}(V)$, for then $f^{-1}(V)$ would be the union of the open neighborhoods $N_x$, $x\in f^{-1}(V)$, and so be open.

Let $𝜀>0$ be small enough so that the neighborhood $M_y$ of $y=f(x)$ of radius $𝜀$ is contained in $V$. Let $n$ be large enough so that $d\left(f_n(z),f(z)\right)\le 𝜀∕3$ for all $z\in E$, which is possible by the assumption of uniform convergence. Let $\delta >0$ such that $d\left(f_n(z),f_n(x)\right)<𝜀∕3$ for all $z$ in the neighborhood $N_x$ of $x$ of radius $\delta$, which is possible since $f_n$ is continuous. Then, for all $z\in N_x$,

That is, $f(z)\in M_y\subset V$, or $z\in f^{-1}(V)$, so that $N_x\subset f^{-1}(V)$.

The remaining Theorems, for mappings into $R^k$, are largely corollaries of their scalar counterparts in the text.

Theorem 7.16 Let $\alpha$ be monotonically increasing on $[a,b]$. Suppose $f_n=(f_{n1},\dots ,f_{\mathit{nk}})\in R(\alpha )$ on $[a,b]$, for $n=1,2,3,\dots ,$ and suppose $f_n\to f=(f_1,\dots ,f_k)$ uniformly on $[a,b]$. Then $f\in R(\alpha )$ on $[a,b]$, and

Theorem 7.17 Suppose $\{f_n=(f_{n1},\dots ,f_{\mathit{nk}})\}$ is a sequence of vector-valued functions, differentiable on $[a,b]$ and such that $\{f(x_0)\}$ converges for some point $x_0$ on $[a,b]$. If $\{f_n^{\prime }\}$ converges uniformly on $[a,b]$, to a function $f=(f_1,\dots ,f_k)$, and

Proofs: If $x=(x_1,\dots ,x_k)\in R^k$, then $|x_i|\le ||x||\le {\sum <!--\; FUNCTION\; APPLICATION\; -->}_i^k|x_i|$. Hence $f_n$ convergences uniformly if and only if the component functions $f_{n1},\dots ,f_{\mathit{nk}}$ converge uniformly. And since the integers or derivatives, and integrability or differentiability, of vector-valued functions are defined by component, the vector-valued versions of the Theorems 7.16 and 7.17 are immediate corollaries of the scalar versions in the text.

Theorem 7.24 If $K$ is a compact metric space, if $f_n=(f_{n1},\dots ,f_{\mathit{nk}})\in C(K)$ for $n=1,2,3,\dots ,$ and if $\{f_n\}$ converges uniformly on $K$, then $\{f_n\}$ is equicontinuous on $K$.

Proof: If $x=(x_1,\dots ,x_k)\in R^k$, then $|x_i|\le ||x||\le {\sum <!--\; FUNCTION\; APPLICATION\; -->}_i|x_i|$. Hence $\{f_n\}$ is equicontinuous if and only if each of the $\{f_{\mathit{ni}}\}$, $i=1,\dots ,k$ are equicontinuous. Hence the vector-valued version of Theorem 7.24 is an immediate corollary of the scalar version in the text.

Theorem 7.25 If $K$ is compact, if $f_n=(f_{n1},\dots ,f_{\mathit{nk}})$ is pointwise bounded and equicontinuous on $K$, then

• $\{f_n\}$ is uniformly bounded on $K$,
• $\{f_n\}$ contains a uniformly convergent subsequence.

Proof: Since each of the sequences $\{f_{\mathit{ni}}\}$, $i=1,\dots ,k$, is pointwise bounded and equicontinuous on $K$, by the scalar version of Theorem 7.25 in the text each of them is uniformly bounded on $K$, so $\{f_n\}$ is uniformly bounded on $K$. Also, $\{f_{n1}\}$ contains a uniformly convergent subsequence, $\{f_{n_{j}1}\}$. Since the subsequence $\{f_{n_{j}2}\}$ is also pointwise bounded and equicontinuous on $K$, it also has a uniformly convergent subsequence such that the corresponding subsequence of $\{f_{n_{j}1}\}$ also converges uniformly. Continuing in this manner, after $k$ steps we have a subsequence $\{f_{n_j}\}$ whose component functions all converge uniformly, so $\{f_{n_j}\}$ also converges uniformly.