Definitions and Theorems,Pre Test # 1:
The absolute value satisfies the following three inequalities: Theorem 1.7
(i)[Positive Definite] For all aЄRwith if and only if a = 0
(ii)[Symmetric] For all a, bЄR,
(iii)The Triangle Inequalities
The Principle of Mathematical Induction (Theorem 1.11)
Suppose for each nЄNthat A(n) is a proposition (a verbal statement or formula) that satisfies the two following properties:
(i)A(1) is true,(A(n) holds for A(1)).
(ii)For every nЄNfor whichA(n) is true, A(n+1) is true also. So assume A(n) is true, then test for A(n+1).
When these two are satisfied, A(n) is true for all nЄN
Definition of Supremum of a set E (Definition 1.16)
Let E R be nonempty.
(i) The set E is said to be bounded above if and only if there is an
MЄRsuch that for all aЄE
(ii) A number M is called an upper bound of the set Eif and only if for all aЄE.
(iii)A number is called a supremum the set E if and only if s is an upperbound of the set E and for all upper bounds M of E(s = sup E).
Definition of Infimum of a set E (Definition 1.27)
Let E R be nonempty.
(i) The set E is said to be bounded below if and only if there is an m Є Rsuch that for all aЄ E.
(ii) A number m is called a lower bound of the set E if and only if for all aЄ E.
(iv)A number is called aninfimum of the set E if and only if t is a lower bound of the set E and for all lower bounds m of E(t= inf E).
Note: A set is said to be BOUNDED if and only if it is BOUNDED ABOVE AND BELOW!
The Completeness Axiom (Postulate 4)
If E is a nonempty subset of R that is bounded above, then E has a finite supremum.
The Archimedean Principle (Theorem 1.22)
Given positive real numbers a and b, there is anintegernЄ N such that b< na
Definition of a 1-1 onto Functions (Definition 1.30)
Let f be a function from a set x into a set y.
(i)f is said to be one to one(an injection) on x if and only if:
x1, x2 Є X and f(x1) = f(x2) imply x1 = x2
(ii)f is said to take x onto y (a surjection) if and only if for each yЄ Y, there is axЄ Xsuch that y = f(x)
Definition of (i) Union, (ii)Intersection of Sets (Definition 1.40)
Let E = {Eά)ά ЄAbe a collection of sets.
(i)the union of the collection E is the set
Eά: = {x : x Є Eά for some ά Є A}
Definition FROM CLASS key word OR
(ii)The intersection of the collection E is the set
Eά: = {x : x Є Eά for some ά Є A}
Definition FROM CLASS key word AND
Definition of Image Inverse Image of a Set under a Function (Definition 1.42)
Let X and Y be sets and f : X → Y.
The image of a set EX under f is the set:
f(E) : = {y Є Y : y = f(x) for some x Є E}
The inverse image of a set EY under f is the set:
f-1(E) : = {x Є X : f(x) = y for some y Є E}
Theorem 1.43(i)(iii)(v)
(i) If {E ά) ά ЄAis a collection of subsets of X, then
and
(i) Alternate NOTES FROM CLASS say: if then
and,
(iii) If {E ά) ά ЄAis a collection of subsets of X, then
and
(iii) Alternate NOTES FROM CLASS say: if then
and,
(v)If , then , but if then
(v) Alternate NOTES FROM CLASS say: if then
And if, then
Definition of Limit of a Sequence (Definition 2.1)
A sequence of real numbers is said to converge to a real number aЄ Rif and only if for every
ε > 0 there is an N Є N ( which in general depends onε) such that:
implies ε
Definition of Subsequence (Definition 2.5)
By a subsequence of a sequence we shall mean a sequence of the form KЄN,
where each nKЄNand n1< n2< n3< n4< …< nknk+1<…
The Squeeze Theorem (Theorem 2.9)
Suppose that {xn},{wn }, {yn}are real sequences
(i)If xn→a and yn→a (the same a) an n → ∞, and if there is NoЄNsuch that:
xn ≤ wn ≤ yn for n ≥ No, then wn→a and n → ∞,
(ii) Ifxn→0as n → ∞, andynis bounded, then xnyn→0as n → ∞
Alternate PROOF from class: The goal isε for alln≥ N
True is ε for alln≥ Nimplies ε – a,and
ε for alln≥ Nimplies ε + a
Pick Nw ≥ Nx, Ny then a - ε >wnε + a
i.e.εfor all n≥ N sot that wn→a and n → ∞
Theorem 2.12 Є
Suppose that {xn} and{yn}are real sequences and ά Є R. If{xn}and{yn}are convergent, then:
(i)
(ii)
(iii)
(v) provided yn ≠ 0
Definition of a Sequence Diverging to Infinity (Definition 2.14)
Let {xn},be a sequence of real numbers
(i) {xn}is said to diverge to +∞ if and only if for each MЄR there is NЄNsuch that:
implies
(ii) {xn}is said to diverge to -∞ if and only if for each MЄR there is NЄNsuch that:
implies
Theorem 2.15
Suppose that {xn} and{yn}are real sequences such that xn→+∞ (respectively xn→-∞) asn → ∞
(i)If Ynis bounded below (respectively Yn bounded above) then:
(respectively
(ii) If ά > 0, then (respectively
Definition of Increasing & Decreasing Sequences (Definition 2.18)
Let {xn}n Є N,be a sequence of real numbers.
(i){xn} is said to be increasing if x1 ≤ x2 ≤ x3 ≤ x4…, and is said to be
strictly increasing if x1 < x2 < x3 < x4…,
(ii){xn} is said to be decreasing if x1 ≥ x2 ≥ x3 ≥ x4 …, and is said to be
strictly decreasing if x1 > x2 > x3 > x4…,
(iii){xn} is said to be monotone if and only if it is either increasing or decreasing.
The Monotone Convergence Theorem (Theorem 2.19)
If {xn} is increasing and bounded above, or if it is decreasing and bounded below, then {xn} has a finite limit.
The Nested Interval Property (Theorem 2.23)
The Bolzano-Weierstrass Theorem (Theorem 2.26)
Every bounded sequence of real numbers has a convergent subsequence.
Sequence x1 , x2 , x3 , x4,… xn,…(n ЄN)
Subsequencexn1, xn2, xn3, xn4,… xnk,…(k ЄN)
So that 0<k1<k2<k3<k4<… ЄR
Example (-1)nwhich does not converge, but is bounded,-1, 1, -1, 1, -1, 1, -1, 1…
has the subsequences Mk = 2k = x2, x4, x6, x8,… which converges to 1 and Mk-1 = 2k-1 = x1, x3, x5, x7,… which converges to -1