Maths Foundations Mid term

Lemma 1 : (∀ x in F) x * 0 = 0

- Use Distributive Law

- Additive inverse of (x*0) is -(x * 0)

Lemma 2 : (∀ x in F) -x = -1 * x

First step: x + ( -1 * x ) = (x * 1) + ( x * -1 )

with (x * 1) being the multiplicative identity and the second part by commutativity.

- Use distributive law with x * (1 + -1)

- Prove RHS = 0, then add (-x) to both sides.

Lemma 3 : (-1)² = 1

• Distributive Law with a = -1, b = -1, c = 1

Lemma 4 : (∀ x in F) If x does not = 0, then x does not equal -x

• Consider trichotomy, so case 1 (x<0) and case 2 (x>0)

• + x

• Use transitivity x + x < x < 0

Lemma 5 : (∀ x in F) x² >= 0

• Trichotomy. 3 cases: x = 0, x > 0, x < 0

• For case 1 and 2 : *x for both sides.

• For case 3 : add -x, so that you can * -x as -x > 0

Lemma 6 : Archimedian Property v1 (the set of natural numbers are unbounded)

• Proof by Contradiction. Assume N is bounded above.

• So it has an upper bound x, so x-1 is not an upper bound. So can pick n>x-1 to exist. so n+1>x false.

Lemma 7 : Archimedean Property v2. 1/n → 0

• Definition of convergence.

• N>1/ε so 1/N<ε

Lemma 8 : Every sequence contains a monotone subsequence.

• 2 cases: infinitely many peaks & finitely many peaks

• Case 1: infinitely many peaks so a_nk >= a_nk+1 so monotone decreasing

• Case 2 : Let a_k be the last peak, so a_n1 < a_n2 so strictly monotone increasing

Proposition 1 : Every rational number has a finite decimal expansion or an infinite recurring decimal expansion.

• Let p/q be in the rational….

• 10ⁿ¹p / q = a₁ + r₁/q

• remainder = 0 then expansion terminates and is finite

• remainder repeats: all subsequent digits repeat

Proposition 2

Root 2 is not rational.

Proposition 3 : A sequence (a_n) is bounded if and only if (∃C in R)(∀n in N) |a_n| <= C

• => Definition of bounded with L and U, set C = max{ |L|, |U| }.

• <= Set L = -C and U = C

Proposition 4 : If a_n is a convergent sequence, then (a_n) is bounded.

• Set ε = 1 and so a - ε < a_n < a + ε

• Consider terms nefore N₁ so (∀n<= N₁ -1) m <= a_n <= M

• (∃m,M in R)(∀n in N)(min { a-1 , m} <= a_n <= max {a+1, M} , and set the min = L and max = U

Proposition 5 : a_n → a <=> every subsequence a_nk → a

• => Convergence def. All indices are strictly increasing, so n_k >= N so replace a_nk with a_n.

• <= Suppose every subsequence converges to a, set k = nk, so replaces a_nk = a_k

Theorem 1 : Triangle inequality

• Write |x| and |y| using expanded bounds and combine

Theorem 2 : Algebraic Limit Theorem

Algebraic Limit Theorem (a_n → a & b_n → b)

• ca_n → ca

• a_n + b_n → a+b

• a_nb_n → ab

• a_n/b_n → a/b

Theorem 2 : Algebraic Limit Theorem part a (ca_n → ca)

• ε / |c|

• (∀ ε / |c| >0 )… where |a_n - a| < ε / |c|

• Multiply the c to the other side

Theorem 2 : Algebraic Limit Theorem part b (a_n + b_n → a+b)

• ε / 2

• Take N_a and N_b and M = max (N_a, N_b)

• Use triangle inequality

Theorem 2 : Algebraic Limit Theorem part c (a_nb_n → ab)

• Take N_a and N_b and M = max (N_a, N_b)

• |a_nb_n - ab| = |a_nb_n - ab_n + ab_n -ab| < …

• Use b_n is bounded so |b_n| <= C and set C into where that is

• Make each term < ε / 2 so +1 on denominator

Theorem 2 : Algebraic Limit Theorem part d (a_n/b_n → a/b)

• Prove (1/b_n → 1/b) then apply part c

• 0 < C < |b|

• |b_n - b| = |b - b_n| < C

• Try to express b_n in terms of something else

Theorem 3 : Sequence Squeeze Theorem ((∀n in N)(a_n <= x_n <= b_n) and a_n → L and b_n → L then, x_n → L

• Express the expanded L - ε < … < L + ε

• Take N_a and N_b and M = max (N_a, N_b)

• Combine using transitivity

Theorem 4: Order Limit Theorem

1. ((∀n in N)(a_n >= 0) => a >= 0

2. ((∀n in N)(a_n <= b_n) => a<=b

3. ((∀c in R)(((∀n in N) c<=b_n) => c<= b)

Theorem 4: Order Limit Theorem part a : ((∀n in N)(a_n >= 0) => a >= 0

• Proof by contradiction : suppose a < 0

• |a_n - a| < |a|. => a_n < 0

Theorem 4: Order Limit Theorem part b : ((∀n in N)(a_n <= b_n) => a<=b

• Consider the sequence (b_n - a_n) → (b - a)

• Use part a so b - a >= 0

Theorem 4: Order Limit Theorem part c : ((∀c in R)(((∀n in N) c<=b_n) => c<= b)

• Define b_n = c (the constant sequence) and apply part b

Theorem 5 : Monotone Convergence Theorem

Monotone + Bounded => Convergence

Theorem 5 : Monotone Convergence Theorem : if monotone + bounded => Convergence

• 2 Cases: increasing case and decreasing case

E.g. Monotone increasing case

• By def of bounded, there exists sup = s

• s - ε < a_N <= a_n <= s < s + ε by transitivity it converges to the sup

Theorem 6 : Bolzano - Weierstrass Theorem

Every bounded sequence contains a convergent subsequence.

(by Lemme 8 we know every sequence contains a monotone subsequence, so if its bounded, then by monotone convergence theorem, it converges)

Def 1 : Partial Ordering

• Reflexivity : a<=a

• Antisymmetry : a<=b and b<=a, then a=b

• Transitivity : a<=b and b<=c , then a<=c

Def 2 : Total Ordering

If it is a partial ordering and any two elements are comparable ( a<=b or b<=a).

Def 3 : Maximal element

No element x is greater than the maximal element m.

Def 4 : Greatest element

(∀s in S) s<= g and g is in S

Def 5 : Field

A set F with two operations + and *, that satisfy the 11 axioms.

Def 6 : Ordered Field

A field with total ordering <=

01) If x <= y , then x+z <= y+z

02) If x <= y , then x*z <= y*z

Def 7 : Upper Bound

(∀s in S) s<= B

Def 8 : Supremum (least upper bound)

1) (∀s in S) s<= B

B is an upper bound

2) (∀y < B)(∃a in S) a>y

No element less than B is an upper bound

Def 9 : Lower Bound

(∀s in S) s >= b

Def 10 : Infimum (greatest lower bound)

1) (∀s in S) s>= b

b is a lower bound

2) (∀y > b)(∃a in S) a<y

No element greater than b is lower bound

Def 11 : Least Upper Bound Property

• If a subset is non empty and is bounded above, then there exists a supremum in the set.

Def 12 : Greatest Lower Bound Property

• If a subset is non empty and is bounded below, then there exists an infimum in the set.

Def 13 : Real Numbers

An ordered field with the least upper bound property.

Def 14 : Absolute value

• |x| = x , if x>=0

• |x|= -x , if x<0

Def 15 : Sequence

A function a : N → R

Def 16 : Convergence of a_n → L

(∀ε >0)(∃N in N)(∀n>=N) |a_n - L| < ε

Def 17 : ε neighbourhood

For a in R & ε >0. Vε (a) = { x in R : |x - a| < ε}

(points whose distance from a is less than ε)

Def 18 : Convergence (topological)

a_n → a if every ε neighbourhood of a contains all terms of the sequence from some point on. Contains finite terms.

(∀ε >0)(∃N in N)(∀n>=N) a_n is in Vε (a)

Def 19 : Bounded

(a_n) is bounded if (∃L,U in R)(∀n in N) L <= a_n <= U

Def 20 : Monotone…

• Monotone increasing if (∀n in N) a_n <= a_n+1

• Strictly monotone decreasing is (∀n in N) a_n > a_n+1

Def 21 : Partial Sum and Series

The m^th partial sum of (b_n) is = b₁ + b₂ + b₃ …

Series is the sequence of partial sums. ie. b₁ , b₁+b₂ , b₁+b₂+b₃, …

If the partial sum converges to B, then the series equals B.

Def 22 : Subsequence

A subsequence of (a_n) is (a_nk). Where the index are strictly monotone increasing.

Def 23 : Peak

a_m is a peak of a sequence (a_n) if (∀n > m) a_m >= a_n