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Definition
The following are equivalent: 1)X is countable 2)There is a surjection ℕ>X 3) There is an injection X>ℕ 


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Definition
Let S be nonempty. 1) An Upperbound for S is u∈ℝ s.t. s≤u ∀s∈S If S has an upperbound, say S is bounded above.
2) A lowerbound for S is l∈ℝ s.t. l≤s ∀s∈S If S has a lower bound, say S is bounded below
S is bounded if bounded above and below 


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Definition
Let S be nonempty. Say x is a supremum (least upper bound) if 1)x is an upperbound 2)if u is another upperbound, x≤u
Say y is an infimum (greatest lower bound) if 1)y is a lower bound 2)if l is another lowerbound, y≥l 


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Definition
Let S be nonempty. Then u=supS iff 1)u≥s s∈S 2)If v less than u then s∈S s.t. s>v 


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Definition
Let S be nonempty. Then u=supS⇔∀ε>0 there is some s∈S s.t. s>uε 


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Definition
If ∅≠S⊆ℝ and S is bounded above the supS exists in ℝ. 


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Thm: Nested Intervals Property 

Definition
Let I_{1}⊇I_{2}⊇I_{3}... be nested, nonempty, closed intervals.
Then
1)∩_{n}I_{n}≠∅
2) If I_{n}=[a_{n},b_{n}] and inf_{n}{b_{n}a_{n}}=0, Then ∩_{n}I_{n} contains a single point 


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Definition
A seq (x_{n}) converges to a∈ℝ if for every ε>0, there is K_{ε}∈ℕ s.t. ∀n>K_{ε}, x_{n}a<ε
We say seq converges if it converges to some a.
If seq doesnt converge to any a, then it diverges
Alt: lim(n→oo) x_{n}a 


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Definition
If (x_{n}) converges, then it is bounded. 


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Definition
Let x_{n}>0 ∀n
Suppose lim x_{n}+1/x_{n}=L<1
Then x_{n}→0 


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Definition
Say (Xn) is Cauchy if for every E>0 there exists k s.t. for every n,m >/k then XnXm 


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Definition
Let r be in R. Xn=r^n Sn=1+r+r^2+... = [1r^(n+1)]/[1r]
If r<1 then r^n>0 so series converges to 1/(1r)
If r>1 then r^n diverges
If r=1 then (Sn) is (1,0,1,0,1,...) diverges
r=/=1 


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Thm: Cluster pt and convergence 

Definition
c is a cluster point of a set A iff there is a seq (Xn) in A converging to c 


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Definition
Let f:A>R and c be a cluster pt of A.
Say L is the limit of f at c if for every E>0, there exists a Delta s.t. x in VsubDelta(c)IntersectionA\{c}, then f(x) is in VsubE(c).
For x in A, cDeltac of f(x) may exist even if f(x) is not defined at x=c
limits are unique when they exist 


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Thm: Sequential Criterion for Limits 

Definition
Let A be contained in F, f:A>R, c is a cluster pt. Then limx>c of f(x)=L iff for EVERY seq (Xn) in A\{c} that converges to c, then f(Xn)>L. 


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Definition
Let A be contained in R and f:A>R and c in A. Say f is continuous at c if either 1) c is a cluster pt of A and limx>c of f(x) exists and is equal to f(c) 2) c is not a cluster pt
By def, f is automatically continous at any isolated pt. f is Discontinuous at c if not cont at c. f is cont on A if cont at all c in A 


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Thm: Equivalent Statements of cont f 

Definition
Let A be contained in R, f:A>R a function, c in A. Then the following are equivalent.
1)f is cont at c
2)For every E>0, there exists Delta>0 s.t. if xc0, there exists Delta>0 s.t. if x is in VsubE(c)IntersectionA then f(x) is in VsubE(f(c))
<> f(VsubDelta(c)IntersectionA) is contained in VsubE(f(c))
<>VsubDelta(c)IntersectionA contained in f^1(VsubE(f(c)) 


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Thm: Continuous Functions on (closed, bounded) intervals 

Definition
Let f:I>R be cont where I is a closed and bdd interval. Then f is bdd.
Note: Result if false if I is not closed or bdd 


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Definition
Let f:I=[a,b]>R be cont. Then there exists a pt c in I s.t. f(c)=supf(I)
There exists some point d in I s.t. f(d)=inff(I)
Say that f attains the max on I 


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Thm: Intermediate Value Theorem 

Definition
Let f:I>R be cont. Suppose a,b are in f(I) and f(a)Then there exists a pt c in I s.t. f(c)=z. 


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Def: Uniformly Continuous 

Definition
We say that f:A>R is uniformly cont on A if for every E>0, there exists Delta>0 s.t. if xy 


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Thm: Uniform Continuity Theorem 

Definition
If f is cont on [a,b], then it is uniformly cont on [a,b] 


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Thm: Continuous Extension Theorem 

Definition
Let f be cont on (a,b). Then f is uniform cont on (a,b) iff there exists a continuous extension g(x) on [a,b]
(If we restrict g(x) to (a,b), it is f(x)) 


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Lemma: Uniform and Cauchy 

Definition
If f is uniform cont on A and (Xn) is Cauchy in A, then f(Xn) is Cauchy 


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Definition
f:J>ℝ is diff at c∈J iff ∃φ:I>ℝ cont at c and f(x)f(c)=φ(x)(xc).
Then f'(c)=φ(c) 


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Thm: Diff and Inverse functions 

Definition
Let I∈ℝ and let f:I>ℝ be strictly monotone and cont on I.
If f is diff at c on I and f'(c)≠0, then f^{1} is diff at d:=f(c) and (f^{1})'(d)=1/[f'(f^{1}(d))] 


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Definition
Let f:[a,b]>ℝ be cont and diff on (a,b). Then ∃c∈(a,b) s.t. f'(c)= [f(b)f(a)]/[ba] 


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Definition
Given f:I>ℝ and a tagged partition P., define the Riemann Sum of f (using P) to be S(f,P.)= ∑(from i=1 to n) f(ti)(xix(i1))
Where f(ti) is the height of the rect. 


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Definition
Say that f:I>ℝ is (Riemann) integrable if ∃L∈ℝ and ∀ε>0, ∃δ>0 such that ∀P satisfying P<δ, then S(f,P)L<ε
Think of this as saying limP>0 of S(f,P)=L
If so, write ∫a to b f(x)dx=L
Riemann Integrable = R 


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Definition
If f is unbdd on [a,b], then f∉R[a,b] 


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Thm: Cauchy Criterion for Integration 

Definition
f:[a,b]>ℝ iff
∀ε>0, ∃γ s.t. if P<γ and Q<γ then S(f,P)S(f,Q)<ε 


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Definition
Let f:[a,b]>ℝ. Say F:[a,b]>ℝ is an anti derivative (primitive) for f if F'(x)=f(x) 


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Definition
if f∈R[a,b] and F is a primitive for f on [a,b].
Then ∫a to b f(x)dx=F(b)F(a) 


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Definition
If f is cont on [a,b] then F(z) is diff on [a,b] AND F'(z)=f(z) 


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Definition
Let {f_{n}} be a seq of functions A→ℝ
Let f:A→ℝ
Say f_{n}→f if f_{n}(a)→f(a) ∀a∈A
This is pointwise convergence 


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Definition
∀a∈A, ∀ε<0, ∃N s.t. n≤N, f_{n}(a)f(a)<ε
Note: N depends on a 


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Definition
∀ε<0, ∃N s.t. n≤N, f_{n}(a)f(a)<ε for every a in A.
Now N does not depend on a. 


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Definition
Let f_{n}:A→ℝ be cont. Suppose f_{n}→f uniformly. Then f is also cont. 


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Definition
Let f_{n}∈R[a,b] and f_{n}→f uniformly. Then f∈R[a,b] and ∫a to b f= lim ∫a to b f_{n} 

