Real analysis/ group theory

Question 1

Trichotomy Property
If x,y are reals, then exactly one of the following three statements are true:

Answer 1

x<y , x=y ,x>y

Question 2

Transitive Property
if a,b,c are reals, then:

Answer 2

if a<b and b<c then a<c

Question 3

Archimedean Property
if a is a real:

Answer 3

then there is a positive integer n such that n is greater than a

Question 4

distance position lemma
|y − b| ≤ a

Answer 4

b − a ≤ y ≤ b + a

Question 5

The Triangle Inequality
For all x, y ∈ R

Answer 5

|x + y| ≤ |x| + |y|

Question 6

The Triangle Inequality
For all a, b, c ∈ R

Answer 6

|a − c| ≤ |a − b| + |b − c|

Question 7

Reverse Triangle Inequality

Answer 7

|c − d| ≥ ||c| − |d||

Question 8

I = (0, 7)

Answer 8

all the values of x which lie between 0 and 7 on the real number line

Question 9

(−3, 2]

Answer 9

x ∈ R with −3 < x ≤ 2

Question 10

Let S be a set of real numbers and let M be a real number. Then M is the maximum of S if…

Answer 10

M ∈ S and for all s ∈ S we have s ≤ M

Question 11

Upper and lower bounds

Answer 11

Since some sets do not have maximums or minimums, we also need to consider the notion of an
upper bound and a lower bound

Question 12

If S is a set of real numbers and b ∈ R we say that b is an upper bound for S if

Answer 12

for all s ∈ S we have s ≤ b

Question 13

If S is a set of real numbers and c ∈ R we say that c is a lower bound for S if

Answer 13

for all s ∈ S we have c ≤ s

Question 14

A subset S of R is said to be bounded above/below if

Answer 14

there exists an upper bound for S
bounded below if there exists a lower bound for S
bounded if it is bounded
above and below

Question 15

We can write the notion of an upper bound using the quantifier ∀
the statement “b is an upper bound for S” is written as

Answer 15

(∀x ∈ S)[x ≤ b]

Question 16

Suppose S ⊆ R is a non-empty subset of R. We say that B ∈ R is the least upper bound (or supremum) for S if:

Answer 16

B is an upper bound for S
if b s an upper bound for S then B≤ b
we write supremum as sup(S) here

Question 16

Suppose S ⊆ R is a non-empty subset of R. We say that C ∈ R is the greatest lower bound (or infinum) for S if:

Answer 16

C is a lower bound for S
if c is a lower bound for S then c ≤ C
we write infinum as inf(S) here

Question 17

Completeness of the real numbers

Answer 17

If S ⊆ R is non-empty and bounded above,
then S has a least upper bound. Similarly, if S ⊆ R is non-empty and bounded below, then S
has a greatest lower bound.

Question 18

Revisit Archimedes

Answer 18

the set of Natural Numbers is not bounded above
If x ∈ R then there is some n ∈ N such that n > x
If y > 0 and z ∈ R, then there is some n ∈ N such that ny > z

Question 19

Corollary 5. For all x ∈ R there exists exactly one integer n such that…

Answer 19

n ≤ x < n + 1
this number is called the integer part of x and is denoted n = ⌊x⌋. In other words, ⌊x⌋ is x rounded down to an integer.

Question 20

sequence

Answer 20

an unending list of real numbers

Question 21

a_n

Answer 21

nth term of the sequence, and the sequence is denoted by (a_n).
We sometimes write (a_n)n∈N

Question 22

Monotonic sequences
Let (a_n) be a sequence of real numbers

Answer 22

(a_n) is increasing if for all n ∈ N we have a_n+1 ≥ a_n
(an) is decreasing if for all n ∈ N we have an+1 ≤ an.
(an) is strictly increasing if for all n ∈ N we have an+1 > an.
(an) is strictly decreasing if for all n ∈ N we have an+1 < an.
(an) is constant if for all n ∈ N we have an+1 = an

Question 23

Sequences and their eventual behaviour
Let (an) be a sequence of real numbers.

Answer 23

(a_n) is constant of value λ if for all n ∈ N we have a_n = λ. It is eventually constant of
value λ if there is N ∈ N such that for all n > N we have an = λ.
* (an) stays less than ε from λ if for all n ∈ N we have |an − λ| < ε. It eventually stays less
than ε from λ if there is N ∈ N such that for all n > N we have |an − λ| < ε

Question 24

(an) is constant of value λ is written as

Answer 24

(∀n ∈ N)[an = λ]

Question 25

(an) is eventually constant of value λ is written as

Answer 25

(∃N ∈ N)(∀n > N )[an = λ]

Question 26

(an) stays less than a distance ε from λ is written

Answer 26

(∀n ∈ N)[|an − λ| < ε].

Question 27

(an) eventually stays less than a distance ε from λ is written

Answer 27

(∃N ∈ N)(∀n > N )[|an − λ| < ε]

Question 28

(Sequence convergence). Let (an) be a sequence of real numbers, and let λ ∈ R.

Answer 28

(an) converges to the real number λ if, given any ε > 0, it eventually stays within ε of λ. In other words, given any ε > 0, there exists a natural number N such that, for all n > N , the distance |an − λ| is less than ε

Question 29

definition of convergence to the limit λ in symbols

Answer 29

(∀ε > 0)(∃N ∈ N)(∀n > N )[|an − λ| < ε]

Question 30

a sequence (an) of real numbers is convergent if there is a real number λ such that (an) converges to λ. We can write the definition of (an) being a convergent sequence in symbols as

Answer 30

(∃λ ∈ R)(∀ε > 0)(∃N ∈ N)(∀n > N )[|an − λ| < ε]

Question 31

If a sequence (an) converges then its limit is

Answer 31

unique

Question 32

(an) does not converge to λ in symbols

Answer 32

(∃ε > 0)(∀N ∈ N)(∃n > N )[|an − λ| ≥ ε] In words, this says that for some positive number ε, however large a natural number N we choose, there is a larger n such that an is still a distance at least ε away from λ

Question 33

Let (an) be a sequence of real numbers. Then:

Answer 33

* (an) is bounded above by M ∈ R if for all n ∈ N we have an ≤ M . * (an) is bounded below by m ∈ R if for all n ∈ N we have m ≤ an. * (an) is eventually bounded above by M ∈ R if for some N ∈ N, whenever n > N we have an ≤ M . * (an) is eventually bounded below by m ∈ R if for some N ∈ N, whenever n > N we have m ≤ an. * (an) diverges to (plus) infinity if for every M ∈ R, there is N ∈ N such that whenever n > N we have an ≥ M .

Question 34

(Algebra of convergent sequences). Suppose we have convergent sequences (an), with limit λ and (bn), with limit μ. Then:

Answer 34

(i) Sum Rule (an + bn) converges to λ + μ (ii) Multiple Rule (ran) converges to rλ, for r ∈ R (iii) Product Rule (anbn) converges to λμ (iv) Quotient Rule an/bn converges to λ/μ provided μ̸ is non zero

Question 35

Let r ∈ R and consider the geometric sequence (rn)

Answer 35

1. If |r| < 1 then limn→∞ rn = 0. 2. If |r| > 1 then (rn) is unbounded. 3. If r = −1 then (rn) is bounded but does not converge

Question 36

(The sandwich principle) If (an), (bn) and (cn) are sequences such that 1. bn ≤ an ≤ cn for all n ∈ N, 2. bn → λ and cn → λ as n → ∞

Answer 36

Then (an) is convergent and furthermore an → λ as n → ∞

Question 37

(Reciprocal Rule). If the sequence (an) satisfies the conditions 1. (an) is eventually positive. 2. 1/an converges with limit 0

Answer 37

then an → ∞

Question 38

(Algebra of divergent sequences). Suppose we have divergent sequences (an) and (bn), both tending to infinity. Then:

Answer 38

(i) Sum Rule (an + bn) → ∞ (ii) Multiple Rule (ran) → ∞, for r ∈ R+ (iii) Product Rule (anbn) → ∞

Question 39

If (bn) tends to infinity and an ≥ bn for all n ∈ N, then

Answer 39

(an) tends to infinity

Question 40

Monotone sequences

Answer 40

A sequence (an) is increasing if an ≤ an+1, for all n ≥ 1; a sequence (bn) is decreasing if bn ≥ bn+1, for all n ≥ 1. A sequence (an) is monotone if it is either increasing or decreasing

Question 41

(The monotone convergence theorem). If (an) be an increasing sequence which is bounded above;

Answer 41

then (an) converges. Similarly, a decreasing sequence which is bounded below also converges.

Question 42

:The sequence (an) given by an = (1+1/n)^n is:

Answer 42

convergent, and its limit e saatisfies 2