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Union (A∪B)
Elements in A or B or both (all)
A∪B = {x: x∈A or x∈B}
Intersection (A∩B)
Elements in both A and B
A∩B = {x: x∈A and x∈B}
Complement of B relative to A (A\B)
Elements in A that are not in B
A\B = {x: x∈A and x∉B}
DeMorgan’s Laws (Theorem 1
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4):
A\(B∪C) = (A\B) ∩ (A\C)
A\(B∩C) = (A\B) ∪ (A\C)
To show that 2 sets are equal, show that they are both contained in each other
Cartesian Product (Definition 1
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5)
Given 2 nonempty sets A and B, A x B = {(a, b): a∈A and b∈B}
(a, b) is an ordered pair
Function (Definition 1
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6)
Let A and B be sets
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When f: A → B, A is the domain and B is the codomain
In other words: If (a, b) ∈ f and (a, b’) ∈ f then b = b’
Function (Definition 1
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6)
Let A and B be sets
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Injective (Definition 1
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9)
A function f: A → B is injective (one-to-one) if whenever 𝑥1 ≠ 𝑥2 then f(𝑥1 ) ≠f(𝑥2 )
Contrapositive: whenever f(𝑥1 ) =f(𝑥2 ) then 𝑥1 = 𝑥2
Surjective (Definition 1
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9)
A function f: A → B is surjective (onto) if for every b∈B there exists a∈A such that f(a) = b
In other words: f(A) = B, or, R(f) = B
Bijective (Definition 1
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9)
A function f: A → B is bijective if it is both injective and surjective
Direct Image (Definition 1
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7)
If E is a subset of A, then the direct image of E under f is the subset f(E) of B given by
f(E):= {f(x) : x ∈ E}
Inverse Image (Definition 1
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7)
If H is a subset of B, then the inverse image of H under f is the subset f^(-1)(H) of A
given

by f^(-1)(H) = {x ∈ A : f(x) ∈ H}
Inverse Function (Definition 1
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11)
If f: A → B is a bijection of A onto B, then f^(-1):= {(b, a) ∈ B x A : (a, b)∈ f}
Composite Function (Definition 1
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12)
If f: A → B and g: B → C, and if R(f) ⊆ D(g) = B, then the composite function g ∘ f is the
function from A to C defined by (g ∘ f)(x):= g(f(x)) for all x∈A
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1
...

Then we have (g ∘ f)^(-1)(H) = f^(-1)(g^(-1)(H))
Well-Ordering Property of N (1
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1)
Every nonempty subset of N has a least element
Principle of Mathematical Induction (1
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2)
Let S be a subset of N with the following properties:
1) 1 ∈S
2) For all k∈N, if k∈S implies k+1∈S, then S is all the natural numbers
Empty Set (Definition 1
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1)
The empty set has zero elements
n elements (Definition 1
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1)
If n∈N, a set S has n elements if there exists a bijection from N onto S
Finite (Definition 1
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1)
A set S is finite if S is empty or has n elements for n∈N
Infinite (Definition 1
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1)
A set S is infinite if it is not finite
Uniqueness Theorem (Theorem 1
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2)
If S is finite, the number of elements is unique
Theorem 1
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3
N is an infinite set
Theorem 1
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4
a) If A and B are disjoint and have m and n elements respectively, then A U B has m+n
elements
b) If A has m elements and C ⊆ A is a set with 1 element, then A\C is a set with m-1
elements
c) If C is an infinite set and B is a finite set, then C\B is an infinite set
Theorem 1
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5
Let S and T be sets such that T ⊆S
If S is finite, then T is finite
If T is infinite, then S is infinite

Denumerable (Definition 1
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6)
A set S is denumerable (countably infinite) if there exists a bijection from N onto S
Countable (Definition 1
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6)
A set S is countable if it is either finite or denumerable
Uncountable (Definition 1
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6)
A set S is uncountable if it is not countable
Some denumerable sets:
N (naturals)
-N
Z (integers)
N x N (theorem 1
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8)
Q (theorem 1
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11)
Theorem 1
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9
Suppose S and T are sets with T⊆S
a) If S is countable, then T is countable
b) If T is uncountable, then S is uncountable
Theorem 1
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10
The following statements are equivalent
a) S is countable
b) There exists a surjection of N onto S
c) There exists an injection of S into N
Theorem 1
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11
The set Q of all rational numbers is denumerable
Theorem 1
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12
If Am is countable for all m ∈ N, then the union from m=1 to infinity of Am is countable
“A countable union of countable sets is countable”
Cantor’s Theorem (Theorem 1
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13)
If A is any set, then there is no surjection of A onto the set P(A) of all subsets of A
Algebraic Properties of R (2
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1)
On the set R of real numbers there are two binary operations, denoted by + and ⋅ and
called addition and multiplication, respectively
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If a0, a1, …, an ∈ Z and r ∈Q satisfy anx^n + … + a1x + a0 = 0 where n≥1, an ≠0, a0
≠0, when we write r = p/q, p, q in Z, with no common factors, then p|a0 and q|an
Theorem 2
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2
(a) If z and a are elements in R with z + a = a, then z = 0
...

(c) If a ∈R, then a ⋅0 = 0
Theorem 2
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3
(a) If a ≠0 and b in R are such that a ⋅b = 1, then b = 1/a
(b) If a ⋅b = 0, then either a = 0 or b = 0
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1
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1
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1
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1
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1
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2
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2
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2
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2
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The 𝜀-neighborhood of a is set V𝜀(a) = {x ∈R: |x - a| < 𝜀}
Theorem 2
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8
Let a be in R
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3
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Any such u is
an upper bound for S
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Any such w is
a lower bound for S
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3
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4
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Corollary 2
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9
If x and y are real numbers with x < y, then there exists an irrational number z such that
xCharacterization Theorem (2
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1)
If S is a subset of R that contains at least two points and has the property
(1) If x, y are in S and x < y, then [x, y] is in S

then S is an interval
Nested Intervals Property (2
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2)
If 𝐼 𝑛 = [𝑎 𝑛 , 𝑏 𝑛 ], n ∈ N, is a nested sequence of closed bounded intervals, then there
exists a number 𝜉 ∈ R such that 𝜉 ∈ 𝐼 𝑛 for all n ∈ N
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5
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1
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1
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The following statements are
equivalent:
(a) X converges to x
(b) For every 𝜀 > 0, there exists a natural number K such that for all n ≥ K, the terms 𝑥 𝑛
satisfy |𝑥 𝑛 - x| < 𝜀
(c) For every 𝜀 > 0, there exists a natural number K such that for all n ≥ K, the terms 𝑥 𝑛
satisfy x - 𝜀 < 𝑥 𝑛 < x + 𝜀
(d) For every 𝜀- neighborhood 𝑉𝜀 (x) of x, there exists a natural number K such that for all
n ≥ K, the terms 𝑥 𝑛 belong to 𝑉𝜀 (x)
m-tail (Definition 3
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8)
If X = (𝑥1 , 𝑥2 , …, 𝑥 𝑛 ,
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1
...
Then the m-tail
𝑋 𝑚 = (𝑥 𝑚+𝑛 : n ∈ N) of X converges if and only if X converges
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1
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If (𝑎 𝑛 ) is a sequence of positive
real numbers with lim(𝑎 𝑛 ) = 0 and if for some constant C > 0 and some m ∈ N we have
|𝑥 𝑛 - x| ≤ C𝑎 𝑛 for all n ≥ m, then it follows that lim(𝑥 𝑛 ) = x
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2
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2
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2
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Then the sequences of X + Y, X - Y, X⋅Y, and cX converge to
x + y, x - y, x ⋅y, and cx, respectively
(b) If X = (𝑥 𝑛 ) converges to x and Z = (𝑧 𝑛 ) is a sequence of nonzero real number that
converges to z and if z ≠0, then the quotient sequence X/Z converges to x/z
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2
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2
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2
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2
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Then Y = (𝑦 𝑛 ) is convergent and
lim(𝑥 𝑛 ) = lim(𝑦 𝑛 ) = lim(𝑧 𝑛 )
Theorem 3
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9
Let the sequence X = (𝑥 𝑛 ) converge to x
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That is, if x = lim(𝑥 𝑛 ), then |x| = lim(|𝑥 𝑛 |)
Theorem 3
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10
Let X = (𝑥 𝑛 ) be a sequence of real numbers that converges to x and suppose that 𝑥 𝑛 ≥
0
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2
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If L <
1,
then (𝑥 𝑛 ) converges and lim(𝑥 𝑛 ) = 0

Definition 3
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1
Let X = (𝑥 𝑛 ) be a sequence of real numbers
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3
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Further:
(a) If X = (𝑥 𝑛 ) is a bounded increasing sequence, then lim(𝑥 𝑛 ) = sup{𝑥 𝑛 : x ∈ N}
(b) If Y = (𝑦 𝑛 ) is a bounded decreasing sequence, then lim(𝑦 𝑛 ) = inf{𝑦 𝑛 : x ∈ N}
Subsequence (Definition 3
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1)
Let X = (𝑥 𝑛 ) be a sequence of real numbers and let 𝑛1 < 𝑛2 < 𝑛3 < … < 𝑛 𝑘 < … be a
strictly increasing sequence of natural numbers
...

Theorem 3
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2
If a sequence X = (𝑥 𝑛 ) of real numbers converges to a real number x, then any
subsequence X’ = (𝑥 𝑛 𝑘 ) of X also converges to x
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4
...
Then the following are equivalent:
(i) The sequence X = (𝑥 𝑛 ) does not converge to x in R
(ii) There exists an 𝜀0 > 0 such that for any k in N, there exists 𝑛 𝑘 in N such that 𝑛 𝑘 ≥ k
and |𝑥 𝑛 𝑘 - x| ≥ 𝜀0
(iii) There exists an 𝜀0 > 0 and a subsequence X’ = (𝑥 𝑛 𝑘 ) of X such that |𝑥 𝑛 𝑘 - x| ≥ 𝜀0 for
All k in N
Divergence Criteria (Theorem 3
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5)
If a sequence X = (𝑥 𝑛 ) of real numbers has either of the following properties, then X is
divergent
(i) X has two convergent subsequences X’ = (𝑥 𝑛 𝑘 ) and X’’ = (𝑥 𝑟 𝑘 ) whose limits are
not equal
(ii) X is unbounded
Monotone Subsequence Theorem (3
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7)
If X = (𝑥 𝑛 ) is a sequence of real numbers, then there is a subsequence of X that is
monotone
The Bolzano-Weierstrass Theorem (3
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8)
A bounded sequence of real numbers has a convergent subsequence
Theorem 3
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9
Let X = (𝑥 𝑛 ) be a bounded sequence of real numbers and let x in R have the property
that every convergent subsequence of X converges to x
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Limit Superior and Inferior (3
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10)
Let X = (𝑥 𝑛 ) be a bounded sequence of real numbers
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An alternative definition of lim sup(xn) (3
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11 (c)):
If 𝑢 𝑚 = sup{𝑥 𝑛 : n ≥ m}, then x* = inf{𝑢 𝑚 : m in N} = lim(𝑢 𝑚 )
Theorem 3
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11
If (𝑥 𝑛 ) is a bounded sequence of real numbers, then the following statements for a real
number x* are equivalent:
(a) x* = lim sup(𝑥 𝑛 )
(b) If 𝜀 > 0, there are at most a finite number of n in N such that x* + 𝜀 < 𝑥 𝑛 , but
an infinite number of n in N such that x* - 𝜀 < 𝑥 𝑛
(c) If 𝑢 𝑚 = sup{𝑥 𝑛 : n ≥ m}, then x* = inf{𝑢 𝑚 : m in N} = lim(𝑢 𝑚 )
(d) If S is the set of subsequential limits of (𝑥 𝑛 ), then x* = sup S
Theorem 3
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12
A bounded sequence (𝑥 𝑛 ) is convergent if and only if lim sup (𝑥 𝑛 ) = lim inf (𝑥 𝑛 )
Cauchy Sequence (Definition 3
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1)
A sequence X = (𝑥 𝑛 ) of real numbers is said to be a Cauchy Sequence if for every 𝜀 > 0
there exists a natural number H(𝜀) such that for all natural number n, m ≥ H(𝜀), the
terms 𝑥 𝑛 , 𝑥 𝑚 satisfy |𝑥 𝑛 - 𝑥 𝑚 | < 𝜀
Lemma 3
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3
If X = (𝑥 𝑛 ) is a convergent sequence of real numbers, then X is a Cauchy sequence
Lemma 3
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4
A Cauchy sequence of real numbers is bounded
Cauchy Convergence Criterion (3
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5)
A sequence of real numbers is convergent if and only if it is a Cauchy sequence
Contractive (definition 3
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7)
We say that a sequence X = (𝑥 𝑛 ) of real numbers is contractive if there exists a constant
C, 0 < C < 1, such that |𝑥 𝑛+2 - 𝑥 𝑛+1 | ≤ C|𝑥 𝑛+1- 𝑥 𝑛 | for all n in N
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Theorem 3
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8
Every contractive sequence is a Cauchy sequence, and therefore is convergent
Definition 3
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1
Let (𝑥 𝑛 ) be a sequence of real numbers
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6
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6
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6
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Then lim(𝑥 𝑛 ) = +∞ if and only if lim(𝑦 𝑛 ) = +∞
Definition 3
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1
If X:= (𝑥 𝑛 ) is a sequence in R, then the infinite series (or simply the series) generated
by
K is the sequence S:= (𝑠 𝑘 ) defined by
𝑠1:= 𝑥1
𝑠2:= 𝑠1 + 𝑥2 = 𝑥1 + 𝑥2
…
𝑠 𝑘 := 𝑠 𝑘−1 + 𝑥 𝑘 = 𝑥1 + 𝑥2 + … + 𝑥 𝑘
…
The nth Term Test (3
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3)
If the series ∑
𝑥 𝑛 converges, then lim(𝑥 𝑛 ) = 0
Cauchy Criterion for Series (3
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4)
The series ∑
𝑥 𝑛 converges if and only if for every 𝜀 > 0 there exists M(𝜀) ∈ N such
that if
m > n ≥ M(𝜀), then |𝑠 𝑚 - 𝑠 𝑛 | = |𝑥 𝑛+1 + 𝑥 𝑛+2 + … + 𝑥 𝑚 | < 𝜀
Theorem 3
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5
Let (𝑥 𝑛 ) be a sequence of nonnegative real numbers
...

In this case ∑∞
𝑥 𝑛 = lim(𝑠 𝑘 ) = sup {𝑠 𝑘 : k in N}
𝑛=1

𝑥𝑛

Comparison Test (3
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7)
Let X:= (𝑥 𝑛 ) and Y:= 𝑦 𝑛 be real sequences and suppose that for some K in N we have
0 ≤ 𝑥 𝑛 ≤ 𝑦 𝑛 for n ≥ K
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7
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1
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A point c ∈ R is a cluster point of A if for every 𝛿 > 0 there exists at least
one point x ∈ A, x ≠c such that |x - c| < 𝛿
Theorem 4
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2
A number c in R is a cluster point of a subset A of R if and only if there exists a
sequence
(𝑎 𝑛 ) in A such that lim (𝑎 𝑛 ) = c and 𝑎 𝑛 ≠c for all n in N
Limit of f at c (4
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4)
Let A ⊆ R, and let c be a cluster point of A
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1
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Theorem 4
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Let f: A → R and let c be a cluster point of A
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1
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Then the following are equivalent
(i) 𝑙𝑖𝑚 𝑓 = L
𝑥→𝑐

(ii) For every sequence (xn) in A that converges to c such that xn ≠c for all n in
N, the sequence (f(xn)) converges to L
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1
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(b) If L in R, then f does not have a limit at c if and only if there exists a sequence
(xn) in A with xn ≠c for all n in N such that the sequence (xn) converges to c but
the sequence (f(xn)) does not converge in R
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2
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We say
that

f is bounded on a neighborhood of c if there exists a delta-neighborhood of c and a
constant M > 0 such that we have |f(x)| ≤ M for all x in A ∩ V(delta)(c)



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Title: Important Definitions & Theorems in Real Analysis (Adv. Calc.)
Description: Key definitions and theorems in analysis. University of Michigan Math 451 (Advanced Calculus).

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