Analysis I

Question 1

Give the axioms of a field

Answer 1

Addition is: unique, commutative & associative and there exists an additive identity & all real numbers have an additive inverse.
Multiplication is: unique, commutative & associative and there exists a multiplicative identity & all real numbers have a multiplicative inverse.
Multiplication distributes over addition
0 ≠ 1

Question 2

Give the axioms of ordering on the real numbers

Answer 2

If a and b are positive, a + b is positive
If a and b are positive, ab is positive
Either a is positive, a=0 or -a is positive

Question 3

Define a > b

Answer 3

a − b ∈ Positive numbers

Question 4

Define a ≥ b

Answer 4

b - a is either positive or 0

Question 5

Define an upper bound

Answer 5

b is an upper bound of S if s ≤ b for all s in S

Question 6

Define a lower bound

Answer 6

b is a lower bound of S if s ≥ b for all s in S

Question 7

Define what it means for a set to be bounded

Answer 7

A set, S is bounded if there exists an upper and a lower bound

Question 8

Define a supremum

Answer 8

Let S be a set. Then α is the supremum of S if:
s ≤ α for all s ∈ S
s ≤ b for all s ∈ S => α ≤ b

Question 9

Give the completeness axiom of the real numbers

Answer 9

Let S be a non-empty subset of R which is bounded above. Then sup S
exists.

Question 10

Define an infimum

Answer 10

Let S be a set. Then α is the infimum of S if:
α ≤ s for all s ∈ S
b ≤ s for all s ∈ S => b ≤ α

Question 11

Define a maximum of a set

Answer 11

For a non empty set, a maximum element max(S) is an element in S such that every element in S is ≤ max(S)

Question 12

Define a minimum of a set

Answer 12

For a non empty set, a minimum element min(S) is an element in S such that every element in S is ≥ min(S)

Question 13

Define |a|

Answer 13

|a| = a if a > 0
|a| = 0 if a = 0
|a| = -a if a < 0

Question 14

Define countably infinitie

Answer 14

A is countably infinite if there exists a bijection between A and the natural numbers

Question 15

Define countable

Answer 15

A is countable if there exists an injection from A to the natural numbers

Question 16

Give the approximation property

Answer 16

Let S be non-empty and bounded above (so sup S exists). Then, given ε > 0, there exists s in S such that: sup S − ε < s < sup S

Question 17

Give the density properties

Answer 17

Given a, b ∈ R with a < b there exists x ∈ Q such that a < x < b.
Given a, b ∈ R with a < b there exists y ∈ R \ Q such that a < y < b.

Question 18

Define a real sequence

Answer 18

A map α:N->R (ie. a sequence is ordered)

Question 19

Define convergence

Answer 19

a_n converges to r if ∀ ε > 0, ∃N in the natural numbers such that |a_n - r|< ε for all n ≥ N

Question 20

Define divergence

Answer 20

a_n is divergent if there is no r which a_n converges to.

Question 21

Define a tail

Answer 21

b_n is a tail of a_n if b_n = a_(n+k) for some k in the natural numbers

Question 22

Give the Sandwich lemma

Answer 22

Let (a_n), (b_n) and (c_n) be real sequences with a_n ≤ b_n ≤ c_n for all n ≥ 1. If a_n → L and c_n → L as n → ∞, then b_n → 0 as n → ∞

Question 23

Define a bounded sequence

Answer 23

If a_n is a sequence, we say that it is bounded if the set {an : n ≥ 1} is bounded.

Question 24

Define a_n → ∞

Answer 24

∀M ∈ R ∃N in the natural numbers such that ∀n ≥ N, a_n > M

Question 25

Define convergence for a complex sequence

Answer 25

z_n -> L means that ∀ε > 0, ∃N in the natural numbers such that ∀n ≥ N, |zn − L| < ε.

Question 26

Define a subsequence

Answer 26

If a_n for n ≥ 0 (natural numbers) is a sequence, b_r, where r ≥ 0 (natural numbers) is a subsequence of a_n provided that there is a function f : n → r such that f is strictly increasing.

Question 27

Define a_n = O(b_n)

Answer 27

If a_n and b_n are sequences, a_n = O(b_n) as n → ∞, means that there is a constant C ∈ R > 0 and N such that if n ≥ N then |an| ≤ C|bn|.

Question 28

Define a_n = o(b_n)

Answer 28

If b_n ≠ 0 for all n (or all sufficiently large n), then a_n = o(b_n) as n → ∞ means that a_n b_n → 0 as n → ∞.

Question 29

Define monotone increasing

Answer 29

a_n is monotone increasing if a_n ≤ a_(n+1) for all n

Question 30

Define strictly increasing

Answer 30

a_n is strictly increasing if a_n < a_(n+1) for all n

Question 31

Define a Cauchy sequence

Answer 31

∀ε > 0, ∃N in the natural numbers such that |a_n - a_m| < ε for all n,m ≥ N

Question 32

Define a rearrangement of a series

Answer 32

Rearranging a series means that we choose a bijection: N -> N that gives a new series

Question 33

Define the exponential function

Answer 33

Sum to infinity of z^k/k!

Question 34

Define the sine function

Answer 34

Sum to infinity of (-1)^k * z^(2k)/(2k+1)!

Question 35

Define the cosine function

Answer 35

Sum to infinity of (-1)^k * z^(2k)/(2k)!

Question 36

Define the radius of convegence

Answer 36

R = sup{|z|:sum of (c_k)(z^k) converges}

Question 37

Define a limit point

Answer 37

p is a limit point if ∀ε > 0, ∃s ≠ p such that 0 < |s-p| < ε