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M1F Foundations of Analysis
Problem Sheet 1
1. Let A be the set {1, 3, −6, {1, −6}, Doncaster, {1}, X}. Which of the following
statements are true and which are false ? (Just write T or F in each case.)
(a) X ∈ A
(b) {X} ∈ A
(c) {X} ̸⊆ A
(d) {1, −6} ∈ A
(e) {1, 3} ̸∈ A
(f) {{1, −6}} ⊆ A
(g) {Doncaster} ⊆ A
(h) {1, −6} ̸⊆ A
(i) ∅ ⊆ A
2. * Describe the following sets. Prove your answers carefully, except in (d) (which
we’ll cover later).
(a)
(b)
(c)
(d)
!∞
n=1 (1/n, ∞)
"∞
n=1 (0, 1/n)
!∞
n=1 {x
"∞
n=1 {x
∈ R: −n < x < n}
∈ Q: 2 −
1
n
< x2 < 2 + n1 }
3. Which of the following statements are true and which are false ?
(a) x2 − 5x + 6 = 0 ⇒ x = 2
(b) x2 − 5x + 6 = 0 ⇐ x = 3
(c) x2 − 5x + 6 = 0 ⇔ (x = 2 or x = 3)
(d) For x2 − 5x + 6 to be zero it is necessary that x = 3
(e) If x2 − 5x + 6 = 0 then x = 3
(f) x = 3 if x2 − 5x + 6 = 0
(g) x = 3 only if x2 − 5x + 6 = 0
(h) x = 1 if x2 − 2x + 1 = 0
4. † Suppose we know that the statement P holds unless Q holds. Which of the
following statements follow and which do not ?
(a) P
(b) Q ⇒ P
(c) Q ⇒ P
(d) Q ⇒ P
You should prepare starred questions * to discuss with your personal tutor.
Questions marked † are slightly harder (closer to exam standard), but good for you.
M1F Foundations of Analysis
Problem Sheet 2
1. What is the biggest element of the set {x ∈ R: x < 1} ? Justify your answer
carefully.
2. Let n be an integer. Prove carefully that if n2 is divisible by 3 then so is n.
(Hint: any integer can be written in the form 3m or 3m + 1 or 3m + 2, for some integer m.)
√
Then prove carefully that 3 is irrational.
3. Are these deductions correct or not ?
(a) My dog barks if I get out of bed on the right. I get out of bed on the left.
Therefore my dog is silent.
(b) My other dog barks only if I get out of bed on the right. I get out of bed
on the left. Therefore he won’t bark.
4. Prove or disprove the following statements:
(a) the sum of two irrational numbers is always irrational
(b) the sum of a rational number and an irrational number is always irrational.
(c) if n and k are positive integers, then nk − n is always divisible by k.
(d) ∃ϵ > 0 such that ∀N ∈ N\{0}, ∀n ≥ N,
1
n
< ϵ.
5. † You throw n infinitely long matches (from your infinitely long matchbox) onto
the ground. Prove that you divide the ground into at most 12 (n2 −3n+2) interior
regions. (You may assume without proof that the earth is flat.)
How can you get equality ?
6. * For which n ∈ N is n! < 2n ?
!
"2
7. Show that 1 + 23 + . . . + n3 = n(n+1)
.
2
You should prepare starred questions * to discuss with your personal tutor.
Questions marked † are slightly harder (closer to exam standard), but good for you.
The reason employers in almost any area will be keen to pay you obscene amounts of money at the
end of your course is because you will have a better grasp of logic and problem solving than other
graduates. To do this well we need the basics to be solid. So we start with basic implications (if,
implies, etc) that may appear obvious but are VERY important and easy to get wrong. If you find
them boring, grit your teeth and think of the cash. If you make small mistakes at this stage, you
won’t be able to solve much harder problems later on.
M1F Foundations of Analysis
Problem Sheet 3
1. Show by induction that 7n − 3n is always divisible by 4.
† Can you see the one-line proof that this true, not using induction ?
2. Write down a careful proof that for any two positive numbers x, y, their mean
(or average, or “arithmetic mean”) 12 (x + y) is greater than or equal to their
√
“geometric mean” xy:
x+y √
≥ xy.
2
When are they equal ?
3. Show that the number of monomials in x, y of degree ≤ k is (k 2 + 3k + 2)/2.
(The monomials of degree k are the polynomials with one term xi y j with i + j = k and
coefficient 1. Here i, j, k are positive but can be 0.)
4. Suppose that an integer n is the sum of two squares (the squares being the
numbers 0, 1, 4, 9, 16, 25, ....). Show that 2n is also.
(There is a small trick involved in solving this; don’t worry if you can’t spot it.)
5. Prove that for every positive integer n ̸= 3, the number
√
n−
√
3 is irrational.
6. * Show that any positive periodic decimal expansion is rational, and in fact can
be written as
!
p 99 . . . 9900 . . . 00 (m 9s and n 0s)
for some integers p, m, n ≥ 0.
Deduce that any integer divides some number of the form 99 . . . 9900 . . . 00.
7. Show that between any two distinct real numbers there exists a rational number
and an irrational number. (For instance you could use decimal expansions.)
You should prepare starred questions * to discuss with your personal tutor.
Questions marked † are slightly harder (closer to exam standard), but good for you.
You need to get good at solving complicated problems that require more than one step. It’s not like
A-level where you could either recognise the answer to a problem immediately, or you couldn’t do it.
You need to be able to tackle problems that you have no idea how to solve at first. Break them
down into little pieces and solve each one. Do (simple) examples to get a feeling for what is true and
why. Draw pictures, do rough working. What makes you more employable than a non-mathematician
is your ability to make progress where others can’t. Don’t just stare at Question 6 and say “I don’t
know how to do this”. Break it down, as we used to say in the ’80s.
M1F Foundations of Analysis
Problem Sheet 4
√
√
1. Irrational Mestel tries to show his tutees that 12 − 3 is rational, by the
following argument.
√
√
12 − 3 = p/q, p, q ∈ N,
√ √
=⇒ 12 − 2 12 3 + 3 = p2 /q 2 ,
√
=⇒ 15 − 2 36 = p2 /q 2 .
√
Since 36 = 6 is indeed rational, this looks good to him. Can you help him by
pointing out three ways in which he’s gone wrong ? Be kind to him!
2. Show that for complex numbers x, y we have x + y = x + y and xy = x.y.
3. * For which z, w ∈ C do we have z + iw = z − iw ?
√
4. (a) What is i ?
√
(b) Find all the 5th roots of 16(1 + 3i).
√
(c) Write (1 + i)(1 − 3i) in the form x + iy and in polar form. Deduce the value
of sin(π/12).
5. Given a complex number u ̸= 1, let z =
1+u
.
1−u
Show that z = −z̄ ⇐⇒ uū = 1.
As u runs round the unit circle, what does z do ?
6. State the fundamental theorem of algebra and factorise
p(z) = z n + a1 z n−1 + . . . + an−1 z + an .
Hence write a1 in terms of the roots of p. Hence give another proof that the roots
of z n = 1 add up to zero when n ≥ 2.
7. Find the solutions of 1 + z + z 2 + . . . + z n−1 = 0.
You should prepare starred questions * to discuss with your personal tutor.
M1F Foundations of Analysis
Problem Sheet 5
1. Professor Mestel is furious. That quadratic equation was on his desk when he
went to lunch, and now he can’t find it anywhere. He can remember its roots –
λ and µ – but not the equation. He tries to cheer himself up by solving another
one,
Ax2 + Bx + C = 0
and is amused to notice that its roots are −(λ + µ)2 and λ2 + µ2 . What was the
equation that he lost ?
2. * Write x2n+1 − 1 as a product of real linear and quadratic factors.
Write x2n + x2n−1 + . . . + x + 1 as a product of quadratic real factors.
! i+j
Suppose that n ≥ 1. Let ω = e2πi/(2n+1) . Why is
ω
= 0, where the sum is
over all i and j from 1 to 2n + 1 such that i < j ?
√
√
3. Imaginary Mestel knows that 1 = 1 and −1 = i. He embarks on a calculation:
"
√
√
−1 = i2 = ( −1)2 = (−1)2 = 1 = 1.
“Liebeck above!” he cries, “M1M1 is not meant to be rigorous, but this is ridiculous!” Where has he gone wrong ?
4. † Suppose that
a + b + c = 3,
a + b2 + c2 = 3,
a3 + b3 + c3 = 3.
2
Find a cubic equation whose roots are a, b, c and hence solve these equations.
5. Find the highest common factor of the numbers 391 and 1904, and express it in
the form 391p + 1904q.
6. What is the highest common factor of 308, 110, 1067 ? Prove it!
You should prepare starred questions * to discuss with your personal tutor.
Questions marked † are slightly harder (closer to exam standard), but good for you.
M1F Foundations of Analysis
Problem Sheet 6
1. In Mestelland all months have exactly 30 days. Much neater. Show that some
months start on a Monday.
2. Professor Mestel likes to have fun every 5 days and eat salad every 8 days. Show
that he sometimes has fun eating salad.
After a fun salad, how long is it until his next fun salad ?
3. The pencils in Professor Mestel’s pencil case have been taken out and laid end
to end, exactly reaching the moon 888 cm away. One is shorter than the others;
the rest are all exactly 18 cm long. What’s the length of the shorter one ?
4. † At home, Professor Mestel keeps an infinite collection of 5 pence and 11 pence
coins. Show that he can buy anything that costs 40 pence or more with the
correct change.
5. ∗ Suppose that m, n are coprime integers that describe some Professor Mestel
related incident, and that m|a and n|a. Show using Euclid’s algorithm that
mn|a.
Show it again using prime factorisation.
Show that it need not be true if m and n are not coprime.
6. † In this question we will give a different proof of the famous consequence of
Euclid’s algorithm from lectures, without using Euclid.
Fix positive integers a and b with highest common factor h := hcf(a, b), and let
S := {x > 0: ∃λ, µ ∈ Z such that x = λa + µb}.
• Show that S is nonempty and bounded below (so it has a minimum element).
• Show that h| min S.
• By dividing min(S) into a and considering the remainder, show that min(S)|a.
• Conclude that h = min S and that ∃λ, µ ∈ Z such that h = λa + µb.
7. Suppose that a and b ̸= 0 are integers. Prove that there exist integers q, r (the
“quotient” and “remainder”) such that a = qb + r where |r| ≤ |b|/2.
Using this to allow negative remainders in Euclid’s algorithm, can you bound
how many steps you need to do to find hcf(a, b) ?
8. Find all solutions of the equation 5n + 2 = m3 (m, n ∈ Z).
You should prepare starred questions * to discuss with your personal tutor.
Questions marked † are slightly harder (closer to exam standard), but good for you.
Always do an example. Draw a picture. Eat a hearty breakfast. i before e except after c.
M1F Foundations of Analysis
Problem Sheet 7
1. Let p be a prime, and fix a which is not equal to 0 or 1 modulo p. Prove that
1 + a + . . . + ap−2 ≡ 0 mod p.
2. (Mini RSA.) Let p be a prime, and fix some e coprime to (p − 1).
(a) Show that there exists d such that de ≡ 1 mod p − 1.
(b) Show that we can solve the equation y ≡ xe mod p by x ≡ y d mod p.
Briefly discuss any relevance for coding.
3. You intercept the brief message “2” encoded by RSA with the woeful public key
(N, e) = (143, 11).
Crack the code and so decode the message.
4. (a) Calculate (p + 1)2 mod p(p + 2).
(b) You notice that your internet bank’s RSA public key is of the form ( N, e) =
(p(p + 2), p) for some pair of prime numbers p, p + 2. The millionaire Martin
Liebeck uses the same bank to save the proceeds from his book, and you
intercept the transmission of his PIN number, which has been encoded as
“p + 1”. Crack the code to find his PIN number.
5. How many numbers between 1 and 30,000,000 are divisible by neither 5 nor 6 ?
6. ∗ Each week Dr McCoy eats 10 packs of McCoy’s Real McCoy crisps, coyly.
A varied diet is important, so she wonders how many weeks she can eat a different distribution1 of packs of her 4 favourite flavours “Normal”, “Poisson”,
“Gamma and pineapple” and “Wallenius’ noncentral univariate hypergeometric
distribution”.
Then she has an idea: she will decide what to eat each week by marking off 3
different numbers from the list {1, 2, . . . , 13}, leaving 10 numbers. The numbers
to the left of the 1st mark will tell her how many packs of Normal to eat, the
numbers between the 1st and 2nd mark will represent packs of Poisson, then
Gamma between the 2nd and 3rd marks, and finally the numbers after the 3rd
mark will denote packets of Wallenius 2 .
How many weeks can she carry on with different distributions before she has to
start eating her least favourite flavour “Student’s tea” ?
† How many ways can you write n ∈ N as a sum n1 + . . . + nk of k nonnegative
integers, where order is 3 important ?
You should prepare starred questions * to discuss with your personal tutor.
Questions marked † are slightly harder (closer to exam standard), but good for you.
For instance last week, she ate 5 packs of Normal, 0 Poisson, 2 Gammas and 3 Wallenius, so the
distribution was 5,0,2,3.
2
So last week the marked numbers were 6,7 and 10.
3
So, for instance, n = a + b and n = b + a count as different ways of expressing n as a sum of two
numbers, unless a = b.
1
M1F Foundations of Analysis
Problem Sheet 8
1. For each relation ∼ below, state which of symmetric, reflexive, transitive and an
equivalence relation it is. ( ⌊x⌋ denotes the integer part of x, i.e. the largest integer n ≤ x.)
(a) On R, x ∼ y ⇐⇒ ⌊x⌋ = ⌊y⌋.
(b) On R, x ∼ y ⇐⇒ ∃n ∈ Z such that x, y ∈ [n − 0.5, n + 0.5).
(c) On R, x ∼ y ⇐⇒ |x − y| < 1.
(d) On Z, x ∼ y ⇐⇒ |x − y| < 1.
2.* (a) How many relations are there on a finite set S ?
(b) How many reflexive relations ?
(c) How many symmetric relations ?
(d) How many symmetric, reflexive relations ?
3. Define a relation on the set Z by x ∼ y if and only if x ≡ y (mod n).
Show this is an equivalence relation. What are the equivalence classes ? How
many are there ?
4. Fix f : S → T and suppose there exists g: T → S such that g ◦ f = idS . Is f
(a) injective ?
(b) surjective ?
(c) bijective ?
Give a proof or counterexample for each.
(You are strongly advised to try to draw a diagram to answer questions like this.)
5. Suppose that S ̸= ∅ and that f : S → T is injective. Prove that there exists
g: T → S such that g ◦ f = idS .
6. Suppose that S, T are finite sets of m, n elements respectively. How many injections are there S → T ? How many bijections ?
7. † We are going to find the number of equivalence relations on the set {1, 2, . . . , n}.
Let this number be Bn (and set B0 = 1). Prove that
! "
! "
n
n
Bn+1 = Bn +
Bn−1 +
Bn−2 + . . . + nB1 + B0 .
1
2
Now let
B(q) =
∞
#
Bn
n!
n=0
qn.
$ $ d
Assuming (without justification) that
=
, that you can reorder infinite
dq
sums, etc., show that
∞
∞
dB # Bk # q n
=
.
dq
k! n=k (n − k)!
k=0
d
dq
Deduce that
dB
dq
= eq B(q). Recalling that B(0) = 1, show that
q −1
B(q) = ee
.
You should prepare starred questions * to discuss with your personal tutor.
Questions marked † are slightly harder (closer to exam standard), but good for you.
M1F Foundations of Analysis
Problem Sheet 9
1. Fix f : S → T and suppose there exists g: T → S such that f ◦ g = idT . Is f
(a) injective ?
(b) surjective ?
(c) bijective ?
Give a proof or counterexample for each.
(You are strongly advised to try to draw a diagram to answer questions like this and Q3.)
2. Consider functions f, g: R → R such that f ◦ g(x) = (x − 1)2 .
(a) If g(x) = x2 , recall from
! lectures
√ that there are no solutions for f . So what is
√
x−2 x+1 x≥0
wrong with f (x) :=
(where x denotes the positive
0
x<0
square root) ?
(b) If f (x) = x2 , then what are the possibilities for g ? How many are there ?
(c) If f (x) = x2 + 2x − 1, then what are the possibilities for g ? How many are
there ?
3. * Suppose given functions h: A → B and g: C → B. Show that there exists
f : A → C such that h = g ◦ f if and only if image (h) ⊆ image (g).
4. † Prove that a function f : B → C is injective if and only if the following statement
holds:
(⋆) for all sets A and all g1 , g2 : A → B, we have f ◦ g1 = f ◦ g2 =⇒ g1 = g2 .
5. Suppose
that each of the sets Sn , n = 1, 2, 3, . . . is countable. Show that S =
"∞
n=1 Sn is also countable.
(Hint: recall the argument used in lectures to show that Q is countable, tracing out
elements along anti-diagonal lines.)
6. Let S 1 = {s11 , s12 , s13 , . . .}, S 2 = {s21 , s22 , s23 , . . .}, . . . , S n = {sn1 , sn2 , sn3 , . . .}, . . . be
subsets of N. Here the elements are ordered so that sni < sni+1 for all i and n.
Define tn recursively to be strictly larger than snn and tn−1 (e.g.
max{snn , tn−1 } + 1).
set tn =
Show that T = {t1 , t2 , . . .} ⊆ N is not equal to any Si . Conclude that the set of
subsets of N is not countable.
7. Suppose
Show that S × T is countable. Hence show
" thatn S and T are countable.
n
that ∞
S
is
countable,
where
S
:=
S × . . . × S (n times).
n=1
8. † Show that the set of polynomials with integer coefficients is countable.
(Hint: use result from previous question.)
A real number is called algebraic if it is a root of a polynomial with integer
coefficients. (For instance all rational numbers are algebraic, as are all square roots of
rational numbers, etc.) Show that the set of algebraic real numbers is countable.
A real number is called transcendental if it is not algebraic. (Examples include π
and e, but this is hard to prove.) Prove that transcendental numbers exist, and
that in fact there are uncountably many of them.
You should prepare starred questions * to discuss with your personal tutor.
Questions marked † are slightly harder (closer to exam standard), but good for you.
M1F Foundations of Analysis
Problem Sheet 10
1. ∗ Fix S ⊂ R with an upper bound, and suppose that S ̸= ∅ and S ̸= R. Give
proofs or counterexamples to the following statements.
(a) If S ⊂ Q then sup S ∈ Q.
(b) If S ⊂ R\Q then sup S ∈ R\Q.
(c) If S ⊂ Z then sup S ∈ Z.
(d) There exists a max S if and only if sup S ∈ S.
(e) sup S = inf(R\S).
(f) sup S = inf(R\S) if and only if S is an interval of the form (−∞, a) or
(−∞, a].
2. Which of the following sequences are convergent and which are not ? What is
the limit of the convergent ones ? Give proofs for each.
(a)
(b)
(c)
n+7
.
n
n
.
n+7
n2 +5n+6
.
n3 −2
(d)
n3 −2
.
n2 +5n+6
(e)
1−n(−1)n
.
n
3. In lectures we defined what it means for a sequence (an ) to converge to a real
number a ∈ R as n → ∞. You and your friends think infinity is cool, so you try
to define what an → +∞ as n → ∞ means.
You come up with the following definitions; which are right and wrong ? (There
might be more than one which is right!) For the wrong ones, illustrate why it is
wrong with an example.
(a) ∀a ∈ R, an ̸→ a.
(b) ∀ϵ > 0 ∃N ∈ N such that n ≥ N ⇒ |an − ∞| < ϵ.
(c) ∀R > 0 ∃N ∈ N such that n ≥ N ⇒ an > R.
(d) ∀a ∈ R ∃ϵ > 0 such that ∀N ∈ N ∃n ≥ N such that |an − a| ≥ ϵ.
(e) ∀ϵ > 0 ∃N ∈ N such that ∀n ≥ N, an > 1ϵ .
(f) ∀n ∈ N, an+1 > an .
(g) ∀R ∈ R, ∃n ∈ N such that an > R.
(h) 1/ max(1, an ) → 0.
4. Let S ⊂ R be nonempty and bounded above. Show that there exists a sequence
of numbers sn ∈ S, n = 1, 2, 3, . . ., such that sn → sup S.
5. † We call a sequence (an ) of real numbers Cauchy if it satisfies the condition:
∀ϵ > 0 ∃N ∈ N such that n, m ≥ N ⇒ |an − am | < ϵ.
We show in lectures that a convergent sequence is Cauchy; we will now show the
converse: that a Cauchy sequence (an ) is convergent.
First show that (an ) is bounded: ∃R ∈ R such that |an | < R ∀n ∈ N.
Let bn = sup{ak : k ≥ n}. Show that bn is also a bounded sequence.
Now let a = inf{bn : n ∈ N}. Prove that an → a as n → ∞.
6. † Special Christmas bonus question. Construction of R from Q.
Slightly abusing the original notation, say that a subset S ⊂ Q is a Dedekind cut
if it satisfies (i) and (ii) below.
(i) If s ∈ S and s > t ∈ Q then t ∈ S (i.e. S is a semi-infinite interval to the left ).
(ii) S has no maximum.
(So once we have the reals we’ll see that the Dedekind cuts are all of the form Sr := (−∞, r)∩Q
for some (any) real number r. But we don’t know what the reals are in this question! )
Then we let R be the set of Dedekind cuts. ( I.e. think of identifying Sr with r ∈ R.)
Check that we can identify Q ⊂ R by taking q ∈ Q to the Dedekind cut Sq :=
{s ∈ Q : s < q}.
Define < on R and show that R has the completeness property: that any bounded
nonempty subset has a least upper bound.
If you’re feeling enthusiastic over Christmas: for two Dedekind cuts S1 , S2 define their sum
S1 + S2 := {s1 + s2 ∈ Q : s1 ∈ S1 , s2 ∈ S2 }. Show that this is also a Dedekind cut. Show that
this operation + on R agrees with the usual + on Q ⊂ R.
Similarly define × on R and / on R\{0} and show they agree with their standard definitions
on Q. Show that +, × satisfy the usual rules of arithmetic (associative, × distributes over +,
0 + x = x and 1 × x = x, etc.).
You should prepare starred questions * to discuss with your personal tutor.
Questions marked † are slightly harder (closer to exam standard), but good for you.