4.12.E: Problems on Sequences and Series of Functions
Complete the proof of Theorems 2 and 3.
Complete the proof of Theorem 4.
In Example \((a),\) show that \(f_{n} \rightarrow+\infty\) (pointwise) on \((1,+\infty),\) but not uniformly so. Prove, however, that the limit is uniform on any interval \([a,+\infty), a>1 .\) (Define "lim \(f_{n}=+\infty\) (uniformly)" in a suitable manner.)
Using Theorem 1, discuss \(\lim _{n \rightarrow \infty} f_{n}\) on \(B\) and \(C(\text { as in Example }(a))\) for each of the following.
(i) \(f_{n}(x)=\frac{x}{n} ; B=E^{1} ; C=[a, b] \subset E^{1}\).
(ii) \(f_{n}(x)=\frac{\cos x+n x}{n} ; B=E^{1}\).
(iii) \(f_{n}(x)=\sum_{k=1}^{n} x^{k} ; B=(-1,1) ; C=[-a, a],|a|<1\).
(iv) \(f_{n}(x)=\frac{x}{1+n x} ; C=[0,+\infty)\).
\(\left.\text { [Hint: Prove that } Q_{n}=\sup \frac{1}{n} (1-\frac{1}{n x+1}\right)=\frac{1}{n} .]\)
(v) \(f_{n}(x)=\cos ^{n} x ; B=\left(0, \frac{\pi}{2}\right), C=\left[\frac{1}{4}, \frac{\pi}{2}\right)\).
(vi) \(f_{n}(x)=\frac{\sin ^{2} n x}{1+n x} ; B=E^{1}\).
(vii) \(f_{n}(x)=\frac{1}{1+x^{n}} ; B=[0,1) ; C=[0, a], 0<a<1\).
Using Theorems 1 and \(2,\) discuss \(\lim f_{n}\) on the sets given below, with
\(f_{n}(x)\) as indicated and \(0<a<+\infty .\) (Calculus rules for maxima and minima are assumed known in (v), \((\mathrm{vi}),\) and (vii).)
(i) \(\frac{n x}{1+n x} ;[a,+\infty),(0, a)\).
(ii) \(\frac{n x}{1+n^{3} x^{3}} ;(a,+\infty),(0, a)\).
(iii) \(\sqrt[n]{\cos x} ;\left(0, \frac{\pi}{2}\right),[0, a], a<\frac{\pi}{2}\).
(iv) \(\frac{x}{n} ;(0, a),(0,+\infty)\).
(v) \(x e^{-n x} ;[0,+\infty) ; E^{1}\).
(vi) \(n x e^{-n x} ;[a,+\infty),(0,+\infty)\).
(vii) \(n x e^{-n x^{2}} ;[a,+\infty),(0,+\infty)\).
[Hint: \(\lim f_{n}\) cannot be uniform if the \(f_{n}\) are continuous on a set, but \(\lim f_{n}\) is not.
[For \((\mathrm{v}), f_{n}\) has a maximum at \(x=\frac{1}{n}\); hence find \(Q_{n}\).]
Define \(f_{n} : E^{1} \rightarrow E^{1}\) by
\[
f_{n}(x)=\left\{\begin{array}{ll}{n x} & {\text { if } 0 \leq x \leq \frac{1}{n}} \\ {2-n x} & {\text { if } \frac{1}{n}<x \leq \frac{2}{n}, \text { and }} \\ {0} & {\text { otherwise }}\end{array}\right.
\]
Show that all \(f_{n}\) and \(\lim f_{n}\) are continuous on each interval \((-a, a),\) \(\left.\text { though } \lim f_{n} \text { exists only pointwise. (Compare this with Theorem } 3 .\right)\)
The function \(f\) found in the proof of Theorem 3 is uniquely determined. Why?
\(\Rightarrow 7.\) Prove that if each of the functions \(f_{n}\) is constant on \(B,\) or if \(B\) is finite, then a pointwise limit of the \(f_{n}\) on \(B\) is also a uniform limit; similarly for series.
\(\Rightarrow 8.\) Prove that if \(f_{n} \rightarrow f(\text { uniformly })\) on \(B\) and if \(C \subseteq B,\) then \(f_{n} \rightarrow f\) (uniformly) on \(C\) as well.
\(\Rightarrow 9.\) Show that if \(f_{n} \rightarrow f(\text { uniformly })\) on each of \(B_{1}, B_{2}, \ldots, B_{m},\) then \(f_{n} \rightarrow f\) (uniformly) on \(\bigcup_{k=1}^{m} B_{k}\).
Disprove it for infinite unions by an example. Do the same for series.
\(\Rightarrow 10.\) Let \(f_{n} \rightarrow f(\text { uniformly })\) on \(B\). Prove the equivalence of the following statements:
(i) Each \(f_{n},\) from a certain \(n\) onward, is bounded on \(B\).
(ii) \(f\) is bounded on \(B\).
(iii) The \(f_{n}\) are ultimately uniformly bounded on \(B ;\) that is, all function values \(f_{n}(x), x \in B,\) from a certain \(n=n_{0}\) onward, are in one and the same globe \(G_{q}(K)\) in the range space.
For real, complex, and vector-valued functions, this means that
\[
\left(\exists K \in E^{1}\right)\left(\forall n \geq n_{0}\right)(\forall x \in B) \quad\left|f_{n}(x)\right|<K .
\]
\(\Rightarrow 11.\) Prove for real, complex, or vector-valued functions \(f_{n}, f, g_{n}, g\) that if
\[
f_{n} \rightarrow f \text { and } g_{n} \rightarrow g \text { (uniformly) on } B ,
\]
then also
\[
f_{n} \pm g_{n} \rightarrow f \pm g(\text { uniformly }) \text { on } B .
\]
\(\Rightarrow 12.\) Prove that if the functions \(f_{n}\) and \(g_{n}\) are real or complex (or if the \(g_{n}\) are vector valued and the \(f_{n}\) are scalar valued), and if
\[
f_{n} \rightarrow f \text { and } g_{n} \rightarrow g \text { (uniformly) on } B ,
\]
then
\[
f_{n} g_{n} \rightarrow f g \text { (uniformly) on } B
\]
provided that either \(f\) and \(g\) or the \(f_{n}\) and \(g_{n}\) are bounded on \(B\) (at least from some \(n\) onward); cf. Problem \(11 .\)
Disprove it for the case where only one of \(f\) and \(g\) is bounded.
[Hint: Let \(f_{n}(x)=x\) and \(g_{n}(x)=1 / n\) (constant) on \(B=E^{1} .\) Give some other examples.]
\(\Rightarrow 13.\) Prove that if \(\left\{f_{n}\right\}\) tends to \(f\) (pointwise or uniformly), so does each subsequence \(\left\{f_{n_{k}}\right\}\).
\(\Rightarrow 14.\) Let the functions \(f_{n}\) and \(g_{n}\) and the constants \(a\) and \(b\) be real or complex \(\left.\text { (or let } a \text { and } b \text { be scalars and } f_{n} \text { and } g_{n} \text { be vector valued }\right) .\) Prove that if
\[
f=\sum_{n=1}^{\infty} f_{n} \text { and } g=\sum_{n=1}^{\infty} g_{n} \text { (pointwise or uniformly) } ,
\]
then
\[
a f+b g=\sum_{n=1}^{\infty}\left(a f_{n}+b g_{n}\right) \text { in the same sense. }
\]
(Infinite limits are excluded.)
In particular,
\[
f \pm g=\sum_{n=1}^{\infty}\left(f_{n} \pm g_{n}\right) \quad \text { (rule of termwise addition) }
\]
and
\[
a f=\sum_{n=1}^{\infty} a f_{n} .
\]
\(\text { [Hint: Use Problems } 11 \text { and } 12 .]\)
\(\Rightarrow 15.\) Let the range space of the functions \(f_{m}\) and \(g\) be \(E^{n}\left(\text {*or } C^{n}\right),\) and let \(f_{m}=\left(f_{m 1}, f_{m 2}, \ldots, f_{m n}\right), g=\left(g_{1}, \ldots, g_{n}\right) ;\) see §3, part II. Prove that
\[
f_{m} \rightarrow g \quad \text { (pointwise or uniformly) }
\]
iff each component \(f_{m k}\) of \(f_{m}\) converges (in the same sense) to the corresponding component \(g_{k}\) of \(g ;\) i.e.,
\[
f_{m k} \rightarrow g_{k} \quad \text { (pointwise or uniformly), } k=1,2, \ldots, n .
\]
Similarly,
\[
g=\sum_{m=1}^{\infty} f_{m}
\]
iff
\[
(\forall k \leq n) \quad g_{k}=\sum_{m=1}^{\infty} f_{m k} .
\]
\(\text { (See Chapter } 3, §15, \text { Theorem } 2)\).
\(\Rightarrow 16.\) From Problem 15 deduce for complex functions that \(f_{m} \rightarrow g\) (pointwise or uniformly) iff the real and imaginary parts of the \(f_{m}\) converge to those of \(g\) (pointwise or uniformly). That is, \(\left(f_{m}\right)_{r e} \rightarrow g_{r e}\) and \(\left(f_{m}\right)_{i m} \rightarrow g_{i m}\); similarly for series.
\(\Rightarrow 17.\) Prove that the convergence or divergence (pointwise or uniformly) of a
sequence \(\left\{f_{m}\right\},\) or a series \(\sum f_{m},\) of functions is not affected by deleting or adding a finite number of terms.
Prove also that \(\lim _{m \rightarrow \infty} f_{m}\) (if any) remains the same, but \(\sum_{m=1}^{\infty} f_{m}\) is altered by the difference between the added and deleted terms.
\(\Rightarrow 18.\) Show that the geometric series with ratio \(r\),
\[
\sum_{n=0}^{\infty} a r^{n} \quad\left(a, r \in E^{1} \text { or } a, r \in C\right) ,
\]
converges iff \(|r|<1,\) in which case
\[
\sum_{n=0}^{\infty} a r^{n}=\frac{a}{1-r}
\]
(similarly if \(a\) is a vector and \(r\) is a scalar). Deduce that \(\sum(-1)^{n}\) diverges. (See Chapter 3, §15, Problem 19.)
Theorem 4 shows that a convergent series does not change its sum if every several consecutive terms are replaced by their sum. Show by an example that the reverse process (splitting each term into several terms) may affect convergence.
[Hint: Consider \(\sum a_{n}\) with \(a_{n}=0 .\) Split \(a_{n}=1-1\) to obtain a divergent series: \(\left.\sum(-1)^{n-1}, \text { with partial sums } 1,0,1,0,1, \ldots\right]\)
Find \(\sum_{n=1}^{\infty} \frac{1}{n(n+1)}\).
\(\left.\text { [Hint: Verify: } \frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1} . \text { Hence find } s_{n}, \text { and let } n \rightarrow \infty .\right]\)
The functions \(f_{n} : A \rightarrow\left(T, \rho^{\prime}\right), A \subseteq(S, \rho)\) are said to be equicontinuous at \(p \in A\) iff
\[
(\forall \varepsilon>0)(\exists \delta>0)(\forall n)\left(\forall x \in A \cap G_{p}(\delta)\right) \quad \rho^{\prime}\left(f_{n}(x), f_{n}(p)\right)<\varepsilon .
\]
Prove that if so, and if \(f_{n} \rightarrow f\) (pointwise) on \(A,\) then \(f\) is continuous at \(p .\)
[Hint: "Imitate" the proof of Theorem 2 .]