500 ALL Flashcards
Bolzano-Weierstrass Theorem (Star theorem)
Every Bounded real sequence has a convergent subsequence
Monotone Convergence Theorem
Every Bounded monotone sequence converges
Cauchy Sequences
if xn is cauchy, then for every e>0, there is an N (natural number) s.t. for all m,n >= N, then |xn - xm| < e (ALL Cauchy sequences are convergent!)
Two-Sided Limit Convergence
Let I be an open interval, a is a point in I, and the function f be defined everywhere on I except possibly at a.
We say that f(x) converges to L as x->a IFF given e>0, there exists d> 0 s.t. for all points x in I, with 0< |x-a|
Equality of limits of functions
Let I be an open interval, a is a point in I, and the functions f,g be defined everywhere on I except possibly at a.
If f(x) = g(x) for all points x in I except a, then lim f(x) = lim g(x)
Sequential Characterization of Limits
Let I be an open interval, a is a point in I, and the function f be defined everywhere on I except possibly at a.
Then lim f(x) = L IFF lim f(xn) = L for all sequences {xn} with xn in I except at a, s.t. lim xn = a.
Comparison Theorem of Limits of Sequences
If {xn} and {yn} converge, then if there exists an N natural number s.t. xn<=yn for all n >=N, then lim(xn) <= lim(yn)
Bad influence Theorem on sequences
if lim(xn) -> 0 and yn is bounded, then lim(xn * yn) -> 0
Squeeze Theorem on Sequences
If {xn} and {yn} both converge to a, and there exists a sequence {zn} s.t. xn < zn < yn, then {zn} -> a
Limit of a Sequence
lim (xn) = a as n -> infinity if given e>0 there exists N a natural number s.t. n>=N implies |xn - a|<e
How many limits can a sequence have
A limit can have at most one limit
Definition of a Subsequence
A subsequence of the sequence {xn} is a sequence that’s obtained by taking a subset of the terms, like {xnk} with nk a natural number
Convergence of subsequences
(for subsequences of a convergent sequence)
if lim(xn) -> a then every subsequence of {xn} also converges to a
Definition of a Sequence
A sequence is a function whose domain is the Natural Numbers . IE it’s a function f: N->R where f(n) = xn is the nth term of the sequence
Two sided Limit and One sided Limits equality
lim f(x) = L as x goes to a IFF lim f(x) x->a+ = lim f(x) x->a- = L
Definition of a right sided limit
f(x) tends to L as x approaches a from the right IFF f is defined on an open interval whose left endpoint is a, and given any e>0 there exists d>0 s.t. for all a<x<a+d we have |f(x)-L|<e
lim f(x) as x->a+ = L
Definition of a left sided limit
f(x) tends to L as x approaches a from the left IFF f is defined on an open interval whose right endpoint is a, and given any e>0 there exists d>0 s.t. for all a-d<x<a we have |f(x)-L|<e
lim f(x) as x->a- = L
Comparison Theorem of limits of functions
Suppose f, g are functions defined on I{a} If lim f(x) and lim g(x) exist, and f(x) <= g(x) for all x in I{a}, then lim f(x) <= lim g(x)
Squeeze Theorem on Functions
Suppose f, g, h defined on I{a} if f(x) <= g(x) <= h(x) for all x in I{a} and lim f(x) = lim h(x) = L, then lim g(x) =L
Bad Influence Theorem on Functions
Let f, g be defined on I{a}. If g is bounded, so |g(x)| <= M for M a real number, and x in I{a} and if lim f(x) = 0, then lim [f(x) * g(x)] = 0
limit of a sequence approaching supremum of a set
Let E be a subset of the real numbers. IF sup(E) < infinity, then there exist a sequence {xn} with elements in E st.t lim xn approaches supE
Definition of Supremum
The supremum of a set E is the smallest upper bound of the set E.
Supremum Approximation Property
If sup(E) is finite, then for all positive e, there exists an element a in set E s.t. a>sup(E)-e
Completness Axiom
If E is a non-empty subset of the real numbers and if E is bounded above, then E has a finite supremum, sup(E) in the real numbers
Archimedian Principle
Given any a, b real numbers with a>0, there exists n a natural number s.t. b<n*a
Density of Rationals
The Rational numbers are dense. For any a,b rational numbers, with a<b, there exists a rational number q s.t. a<q<b
Definition of Infimum
The infimum of a set E is the largest possible lower bound of the set E.
Reflection Principle
Let E be a subset of the real numbers, and not an empty set. 1) E has a supremum iff -E has an infimum, and sup(E) = -inf(E) 2) E has an infimum iff -E has a supremum, and inf(E) is -sup(E)
Monotone Property
Let A be a subset of B, both sets in real numbers, non-empty. Then, 1) if B has a supremum, then sup(A) <= sup(B) 2) if B has an infimum, then inf(B) <= inf(A)
Well-Ordering Principle
If E is a non-empty subset of the NATURAL numbers, then E has a least element
Induction Theorem
For every natural number n, supposed A(n) is a statement where A(1) is true, and for all n where A(n) is true, A(n+1) is also true
Definition of a Set
A set is a collection of elements
Definition of a Subset
A subset B of set A is if all of the elements in B are also in set A. If B is not equal to A, that is a proper subset
A ∩ B
{x: x in A and x in B}
A U B
{x: x in A or x in B}
If E is a collection of sets, we have ∩A
{x: x in A for every A in E}
If E is a collection of sets, we have UA
{x: x in A for some A in E}
If B a subset of A, Complement of B in A
A\B, all of the elements of A not in B
Definition of a function
A function f from set A to set B is a rule that assigns to each element of A exactly one element of B.
Images
f(x) is called the image of x under f
Onto/ Surjective
a function f:A->B is onto if for all elements y in B, there is an x in A where y is an element of f(x) (every point y in B has an associated point x in A)
One-to-One / Injective
a function f:A-> is injective if f(x1)=f(x2) implies x1=x2 for all xn in A (Each x in A only has one corresponding point y in B
Bijection
Both injective and surjective
Definition of Function Inverses
for a function f:A->B has an inverse iff there exists a function g:B->A where g(f(x))=x for all x in A, and f(g(y)) = y for all y in B
Uniqueness of inverses
If a function has an inverse, that inverse is unique
definition of a two-sided limit as x goes to infinity
f(x)->L as x->inf iff there exists c>0 s.t. (c, inf) is in the domain of the function, and given any e>0, there exists an M >= c s.t for all x > M we have |f(x) - L| < e
definition of two-sided limit as x goes to negative infinity
f(x)->L as x->-inf iff there exists c>0 s.t. (-inf, -c) is in the domain of the function, and given any e>0, there exists an M <= c s.t for all x < M we have |f(x) - L| < e
Definition of a two-sided limit as f diverges to infinity
f diverges to infinity as x->a iff given M>0 there exists d>0 s.t. for all |x-a|<d we have f(x) > M
Definition of a two-sided limit as f diverges to negative infinity
f diverges to negative infinity as x->a iff given M<0, there exists d>0 s.t. for all|x-a|<d we have f(x) <M
