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|g(t)| ¤ 1 and |g (t)| ¤ 1 for all t ∈ R and g(0) = 1.
(e) Given M0 , M2 > 0 we can ¬nd an in¬nitely di¬erentiable function
G : R ’ R with |G(t)| ¤ M0 and |G (t)| ¤ M2 for all t ∈ R and G(0) =
(M0 M2 )1/2 .
(f) There exists a constant A such that, if |f (t)| ¤ M0 and |f (t)| ¤ M2
for all t ∈ R, then |f (t)| ¤ A(M0 M2 )1/4 for all t ∈ R.
Exercise K.53. [4.5, P] We consider functions f : (0, 1) ’ R. Give a proof
or a counterexample for each of these statements.
(i) If f is uniformly continuous, then f is bounded.
(ii) If f is uniformly continuous, then f is bounded and attains its bounds.
(iii) If f is continuous and bounded and attains its bounds, then f is
uniformly continuous.
Exercise K.54. [4.5, P] Give a proof or a counterexample for each of these
statements.
(i) If f : R ’ R is continuous and |f (x)| tends to a limit as |x| ’ ∞,
then f is uniformly continuous.
(ii) If f : R ’ R is uniformly continuous, then |f (x)| tends to a limit as
|x| ’ ∞.
459
Please send corrections however trivial to twk@dpmms.cam.ac.uk

(iii) If f : R ’ R is continuous and bounded then f is uniformly contin-
uous.
(iv) If f : R ’ R is uniformly continuous, then f is bounded.
(v) If f : C ’ C is continuous and |f (z)| tends to a limit as |z| ’ ∞,
then f is uniformly continuous.
(vi) If f : C ’ C is uniformly continuous, then so is |f | : C ’ R.
(vii) If f : R ’ C is continuous and |f | : R ’ R is uniformly continuous,
then f is uniformly continuous.
(vii) If f, g : R ’ R are uniformly continuous, then so is their product
f — g.
(viii) If f, g : R ’ R are uniformly continuous, then so is their composi-
tion f —¦ g.

Exercise K.55. [4.5, P] If E is a non-empty subset of Rn we de¬ne f :
Rn ’ R by

f (x) = inf{ x ’ y : y ∈ E}.

Show that f is uniformly continuous.
Show also that E is closed if and only if given x ∈ Rn we can ¬nd y ∈ E
such that x ’ y = f (x).

Exercise K.56. [4.5, P] Suppose that f : Q ’ R is uniformly continuous
on Q. The object of this question is to show that f has a unique continuous
extension to R.
(i) By using the general principle of convergence, show that, if x ∈ R,
xn ∈ Q, [n ≥ 1] and xn ’ x as n ’ ∞, then f (xn ) tends to a limit.
(ii) Show that if x ∈ R, xn , yn ∈ Q [n ≥ 1] and xn ’ x is such that
xn ’ x and yn ’ x, then f (xn ) and f (yn ) tend to the same limit.
(iii) Conclude that there is a unique F : R ’ R de¬ned by f (xn ) ’ F (x)
whenever xn ∈ Q and xn ’ x.
(iv) Explain why F (x) = f (x) whenever x ∈ Q.
(v) Show that F is uniformly continuous.
[See also Exercise K.303.]

Exercise K.57. [4.6, P] Suppose that the power series ∞ an z n has ra-
n=0
∞ n
dius of convergence R and the power series n=0 bn z has radius of conver-
gence S.
(i) Show that, if R = S, then ∞ (an + bn )z n has radius of convergence
n=0
min(R, S).
(ii) Show that, if R = S, then ∞ (an + bn )z n has radius of convergence
n=0
T ≥ R.
460 A COMPANION TO ANALYSIS

(iii) Continuing with the notation of (i) show, by means of examples, that
T can take any value with T ≥ R.
(iv) If » = 0 ¬nd the radius of convergence of ∞ »an z n . What happens
n=0
if » = 0?
(v) Investigate what, if anything, we can say about the radius of con-
∞ n
if |cn | = max(|an |, |bn |). Do the same if |cn | =
vergence of n=0 cn z
min(|an |, |bn |).
Exercise K.58. [4.6, P] We work in C. Consider a series of the form
∞ nz
n=0 bn e . Show that there exists an X, which may be ∞, ’∞ (with
appropriate conventions) or any real number, such that the series converges
whenever z < X and diverges whenever z > X. Show, by means of
examples, that any such value of X may occur.
Find the value of X for the sum

2n enz
.
(n + 1)2
n=0

By using results from the ¬rst paragraph, show that, if ∞ cn converges,
n=0
then there exists a Y , which may be ∞ (with an appropriate convention)
or any positive real number, such that ∞ cn cos nz converges absolutely
n=0
whenever | z| < Y and diverges whenever | z| > Y .
Exercise K.59. [4.6, T] Consider the power series ∞ an z n . Show that,
n=0

1/n n
if the sequence |an | is bounded, then n=0 an z has in¬nite radius of
1/n
convergence if lim supn’∞ |an | = 0, and radius of convergence

R = (lim sup |an |1/n )’1
n’∞

otherwise. What can we say if the sequence |an |1/n is unbounded? Prove
your statement.
This formula for the radius of convergence is of considerable theoretical
importance but is hard to handle as a practical tool. It is usually best to use
the de¬nition directly.
Exercise K.60. [4.6, T] Suppose that the sequence an of strictly positive
real numbers has the property that there exists a K ≥ 1 with K ≥ an+1 /an ≥
K ’1 for all n ≥ 1. Prove that

lim inf (an+1 /an ) ¤ lim inf a1/n ¤ lim sup a1/n ¤ lim sup(an+1 /an ).
n n
n’∞ n’∞ n’∞ n’∞

1/n
Conclude that, if an+1 /an ’ l as n ’ ∞, then an ’ l.
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Give examples of sequences an such that
1/n
(i) an ’ l for some l but lim inf n’∞ (an+1 /an ) < l < lim supn’∞ (an+1 /an ).
1/n
(ii) lim inf n’∞ (an+1 /an ) = lim supn’∞ an < lim supn’∞ (an+1 /an ).
In how many di¬erent ways can you replace the ¤ in the initial formula
by combinations of < and =? Can all these possibilities occur? Justify your
answer.
Comment very brie¬‚y on the connection with formulae for the radius of
convergence of a power series given in Exercise K.59.

Exercise K.61. (Summation methods.) [4.6, T] If bj ∈ R, let us write

b0 + b 1 + · · · + b n
Cn = .
n+1
(i) Let > 0. Show that, if |bj | ¤ for all j ≥ 0, then |Cn | ¤ for all
n ≥ 0.
(ii) Show that, if N ≥ 0 and bj = 0 for all n ≥ N , then Cn ’ 0 as n ’ ∞.
(iii) Show that, if bj ’ 0 as j ’ ∞, then Cn ’ 0 as n ’ ∞.
(iv) By considering bj ’ b, or otherwise, show that, if bj ’ b as j ’ ∞,
then Cn ’ b as n ’ ∞.
(v) Let bj = (’1)j . Show that Cn tends to a limit (to be found) but bj
does not.
(vi) Let b2m +k = (’1)m for 0 ¤ k ¤ 2m ’ 1 and m ≥ 1. Show that Cn
does not tend to a limit as n ’ ∞.
(vii) If 1 > r > 0 and bj ∈ R, let us write

(1 ’ r)r n bn .
Ar =
n=0

Show that, if bj ’ b as j ’ ∞, then Ar ’ b as r ’ 1 through values of
r < 1. Give an example where Ar tends to limit but bj does not. Show that
there exists a sequence bj with |bj | ¤ 1 where Ar does not tend to a limit as
r ’ 1 through values of r < 1.
(viii) Let bj ∈ Rm . Show, that, if bj ’ b as j ’ ∞, then

b0 + b 1 + · · · + b n
’b
n+1
You should give two proofs.
(A) By looking at coordinates and using the result for R.
(B) Directly, not using earlier results.
(ix) Explain brie¬‚y why (vii) can be generalised in the manner of (viii).
462 A COMPANION TO ANALYSIS

Exercise K.62. [4.6, T, ‘ ] We continue with the notation of Exercise K.61.
We call the limit of Cn , if it exists, the Ces`ro limit of the sequence bj . We
a
call the limit of Ar if it exists, the Abel limit of the sequence bj . Now suppose
aj ∈ R and we set bj = j aj . r=0
(i) Show that
n
1
Cn = (n + 1 ’ k)ak
n+1 k=0

and

r k ak .
Ar =
k=0

If Cn tends to a limit, we say that the sequence aj has that limit as a
Ces`ro sum and that the sequence aj is Ces`ro summable. If Ar tends to a
a a
limit we say that the sequence aj has that limit as a Abel sum and that the
sequence aj is Abel summable.
(ii) Explain very brie¬‚y why the results of Exercise K.61 (viii) and (ix)
show that we can extend the de¬nitions of (i) to the case aj ∈ C.
(iii) Let aj = z j with z ∈ C.
(a) Show that ∞ z j converges if and only if |z| < 1 and ¬nd the sum
j=0
when it exists.
(b) Show that the sequence z j is Ces`ro summable if and only if |z| ¤ 1
a
and z = 1. Find the Ces`ro sum when it exists.
a
(c) Show that the sequence z j is Abel summable if and only if |z| ¤ 1
and z = 1. Find the Abel sum when it exists.

Exercise K.63. [4.6, T, ‘ ] (This generalises parts of Exercise K.61) Sup-
pose that ujk ∈ R for j, k ≥ 0. Suppose that
(1) If k is ¬xed, ujk ’ 0 as j ’ ∞.

(2) k=0 ujk is absolutely convergent and there exists a M such that

k=0 |ujk | ¤ M for each j ≥ 0.
(3) ∞ ujk ’ 1 as j ’ ∞.
k=0
(i) Show that, if bj ∈ R and bj ’ b as j ’ ∞, then

ujk bk ’ b
k=0

as j ’ ∞.
(ii) Explain why the result just proved gives part (iv) of Exercise K.61.
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Please send corrections however trivial to twk@dpmms.cam.ac.uk

(iii) Let 0 < rn < 1 and rn ’ 1. Use (i) to show that, in the notation of
Exercise K.61 (vii), if bj ’ b as j ’ ∞, then Arn ’ b as n ’ ∞. Deduce
Exercise K.61 (vii).
(iv) State and prove an extension of part (i) along the lines of Exer-
cise K.61 (viii).
(v) If » > 0 and bj ∈ C, let us write

»n ’»
B» = e bn .
n!
n=0

Show that, if bj ’ b as j ’ ∞, then B» ’ b as » ’ ∞.
(vi) Much as in the introduction to Exercise K.62 we suppose aj ∈ C and
set bj = j aj . Using the notation of part (v), we say that, if B» tends
r=0
to a limit, then the sequence aj has that limit as a Borel sum and that the
sequence aj is Borel summable.
Show that the sequence z j is Borel summable if and only if z < 1. Find
the Borel sum when it exists.

Exercise K.64. [4.6, P, ‘ ] Let us write G for the set of real double se-
quences U = (ujk )j,k≥0 which satisfy conditions (1), (2) and (3) of Exer-
cise K.63. By providing a proof or counterexample as appropriate, establish
which of the following statements are true and which are false.
(i) If bj is a bounded real sequence, there exists a U ∈ G such that

k=0 ujk bk tends to a limit as j ’ ∞. (Hint. Bolzano-Weierstrass.)
(ii) If bj is a bounded real sequence which does not tend to a limit, there
exist U, V ∈ G such that ∞ ujk bk and ∞ vjk bk converge to di¬erent
k=0 k=0
limits as j ’ ∞.
(iii) If bj is a bounded real sequence, we can ¬nd real numbers ± and β
such that there exists a U ∈ G having the property that ∞ ujk bk tends to
k=0
» if and only if » ∈ [±, β].
(iv) If U ∈ G, we can ¬nd a bounded real sequence bj such that ∞ ujk bk
k=0
does not tend to limit as j ’ ∞.
(v) If bj is a bounded real sequence, we can ¬nd a U ∈ G such that

k=0 ujk bk does not tend to limit as j ’ ∞.

Exercise K.65. [4.6, T!, ‘ ] The object of this exercise is to prove the
converse of part (i) of Exercise K.63. In other words we want to show that,
if ujk ∈ R for j, k ≥ 0 and

ujk bk ’ b
k=0
464 A COMPANION TO ANALYSIS

as j ’ ∞ whenever bj ∈ R and bj ’ b as j ’ ∞, then it follows that the
ujk satisfy conditions (1), (2) and (3) of Exercise K.63. (This is quite hard
work and simpler proofs exist using more advanced techniques.)
(i) By choosing particular sequences bk , show that, if the ujk satisfy the
stated hypothesis, then they must satisfy conditions (1) and (3).
(ii) Show that if ∞ ujk bk exists for each convergent sequence bk , then
k=0

k=0 ujk is absolutely convergent. (Look at Exercise 5.1.11 if you need a
hint.)
(iii) Suppose that bk is given for 0 ¤ k ¤ N ’ 1 with |bk | ¤ L Suppose
that · > 0, > 0, |uk | ¤ · for 1 ¤ k ¤ N ’ 1 and M ’1 |uk | ≥ K for some
k=1
M > N . Show that we can ¬nd bk [N ¤ k ¤ M ’ 1] such that |bk | ¤ for
N ¤ k ¤ M ’ 1 and
M ’1
uk bk ≥ (K ’ N ·) ’ N L·.
k=0


(iii) Suppose that the ujk satisfy conditions (1) and (3) together with
the conclusion of (ii), but do not satisfy condition (2). Show, by induction,
or otherwise, that we can ¬nd a sequence of integers 0 = N (0) < N (1) <
N (2) < . . . , a sequence of integers 0 = j(0) < j(1) < (2) < . . . , a sequence
of real numbers 1 = (0) > (1) > (2) > . . . with 2’r ≥ (r) > 0, and a
sequence bk of real numbers such that

|bk | ¤ (r) for N (r) ¤ k ¤ N (r + 1) ’ 1
N (r+1)’1
uj(r)k bk ≥ 2r + 1
k=0

uj(r)k xk ¤ 1 provided |xk | ¤ (r + 1) for all k ≥ N (r + 1).
k=N (r+1)


Show that | ∞ uj(r)k bk | ≥ 2r and use contradiction to deduce the result
k=0
stated at the beginning of the exercise.

Exercise K.66. [5.2, P] We work in Rn . Show that the following state-
ments are equivalent.
(i) ∞ xn converges absolutely.
n=1
(ii) Given any sequence n with n = ±1, ∞ n xn converges.
n=1

Exercise K.67. [5.2, P] Let S1 and S2 be a partition of Z+ . (That is to
say, S1 ∪ S2 = Z+ and S1 © S2 = ….) Show that, if an ≥ 0 for all n ≥ 1, then
465
Please send corrections however trivial to twk@dpmms.cam.ac.uk


an converges if and only if n∈S1 an converges (that is n∈S1 , n¤N an
n=1
tends to a limit as N ’ ∞) and n∈S2 an converges.
Establish, using proofs or counterexamples, which parts of the ¬rst para-
graph remain true and which become false if we drop the condition an ≥ 0.
Show that if an , bn ≥ 0, ∞ an and ∞ bn converge, and max(an , bn ) ≥
n=1 n=1
cn ≥ 0 then ∞ cn converges.
n=1
Show that if an ≥ 0 for all n and ∞ an converges then
n=1

∞ 1/2
an
converges.
n
n=1

Give an alternative proof of this result using the Cauchy-Schwarz inequal-
ity.
Suppose that an , bn ≥ 0, ∞ an and ∞ bn diverge, and max(an , bn ) ¤
n=1 n=1

cn . Does it follow that n=1 cn diverges? Give a proof or a counterexample.

Exercise K.68. (Testing for convergence of sums.)[5.2, P] There is
no certain way for ¬nding out if a sum ∞ an converges or not. However,
n=0
there are a number of steps that you might run through.
(1) Is the matter obvious? If an 0 then the sum cannot converge. Does
the ratio test work?
(2) Are the ¬rst few terms untypical? The convergence or otherwise of
∞ ∞
n=N an determines that of n=1 an .
(3) (Mainly applies to examination questions.) Is the sum ∞ an ob-n=0

tained from a sum n=0 bn whose behaviour is known by adding or omitting
irrelevant terms? (For examples see (i) and (ii) below.)
(4) Check for absolute convergence before checking for convergence. It is
generally easier to investigate absolute convergence than convergence and, of
course, absolute convergence implies convergence.
(5) To test for the convergence of a sum of positive terms (such as arises
when we check for absolute convergence) try to ¬nd another sum whose
behaviour is known and which will allow you to use the comparison test.
(6) To test for the convergence of a sum of decreasing positive terms
consider using the integral test (see Lemma 9.2.4) or the Cauchy condensation
test.
(7) If (5) and (6) fail, try to combine them or use some of the ideas of
parts (iii) and (iv) below. Since all that is needed to ¬nd whether the sum

n=1 an of positive terms converges is to discover whether the partial sums
are bounded (that is, there exists a K with N an ¤ K for all N ) it is
n=1
rarely hard to discover whether a naturally occurring sum of positive terms
converges or not.
466 A COMPANION TO ANALYSIS

(8) If the series is not absolutely convergent you may be able to show
that it is convergent using the alternating series test.
(9) If (8) fails, then the more general Abel™s test (Lemma 5.2.4) may
work.
(10) If (8) and (9) fail then grouping of terms (see part (vi)) may help
but you must be careful (see parts (v) and (vii)).
(11) If none of the above is helpful and you are dealing with a naturally
occurring in¬nite sum which is not absolutely convergent but which you hope
is convergent, remember that the only way that such a series can converge
is by ˜near perfect cancellation of terms™. Is there a natural reason why the
terms should (almost) cancel one another out?
(12) If you reach step (12) console yourself with the thought that where
routine ends, mathematics begins4 .

1 + (’1)n
(i) State, with proof, whether converges.
n
n=1

1
(ii) Let pn be the nth prime. State, with proof, whether , converges.

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