Math 117 Midterm 1 Terms

Flashcards by esnyder1994, updated more than 1 year ago
Created by esnyder1994 over 5 years ago


First half of terms covered by Math 117: Introduction to Analysis at UCSB

Resource summary

Question Answer
Trichotomy For all a, b exactly one of these is true a = b, a < b, or b < a Applies to Q and R
Upper-Bound Given A in R is a non-empty set, b is an upper bound for A if a <= b for all a in A
Supremum (Infimum) s is a supremum (infimum) of A if s is an upper-bound (lower-bound) for A and s<=b (s>=b) for all upper-bounds (lower-bounds) b of A
Axiom of Completeness If A of R is non-empty and bounded above, then sup(A) exists as a real number
Approximation Property Given a non-empty set A of R such that s is an upper bound of A. Then s = sup(A) iff for all e > 0, there is an a in A such that s-e < a
Archimedian Property Given x,y in R, both positive, there is an n in N such that nx > y
Density of Q in R Given a < b where a,b in N, there is an r such that a < r < b and r in R
Equivalence Relation A relation '~' on sets by A ~ B iff there is a bijection A --> B ( 1-to-1 and Onto)
Countable We say an infinite set A is countable if |A| = N_0 i.e. there is a bijection N --> A : n |--> a_n
Contained Countable Sets If A is contained by B and B is countable, then A is either finite or countable
Injection 1-to-1, two unique elements in the domain are never mapped to the same element in the codomain
Cantor's Theorem For any set S, the size of S is less than the power set of S or the set of all subsets of S |S| < |P(S)| where P(S) = {T contained in S}
Metric Space A metric space (M,d) is a set M and a distance function d: M x M --> R such that for all x,y, z in M: 1. d(x,x) = 0 and d(x,y) > 0 if x != y (Positivity) 2. d(x,y) = d(y,x) (Symmetry) 3. d(x,z) <= d(x,y) + d(y,z) (Triangle Ineq.)
Sequence In a metric space (M,d), a sequence is an infinite list: x_1, x_2, x_3 ... i.e. a function N --> M: n |--> x_n Various notations such as (x_n) from 1 to infinity, (x_n), or (x_1, x_2, x_3, ...)
Limit A sequence (x_n) in a metric space (M,d) is said to have a limit L in M if for every e>0 there is an N in N (depending on e) such that n >= N ==> d(x_n, L) < e
Algebraic Properties of Limits of R Sequences Let (a_n), (b_n) be convergent sequences of real numbers with limits a, b respectively 1. lim(c * a_n) = c * a for any c in R 2. lim(a_n + b_n) = a + b 3. lim(a_n * b_n) = a * b
Comparison Theorem Let (a_n), (b_n) be convergent sequences of real numbers with limits a, b respectively. If a_n <= b_n for all n in N, then a <= b
Bounded A sequence (x_n) in a metric space is bounded when there is a ball Br(x) = {m in M, d(m, x) < r} such that x_n is in Br(x) for all n in N
Convergent, Bounded Fact Convergent sequences are bounded, but bounded sequences are not necessarily convergent. Convergent ==> Bounded, Bounded =/=> Convergent
Increasing (Decreasing) Given a sequence (a_n) of real numbers, we say (a_n) is increasing (decreasing) if a_n <= a_n+1 (a_n >= a_n+1) for all n in N (a_n) is monotone if a_n is either increasing or decreasing
Monotone Convergence Theorem (MCT) Given a bounded monotone sequence (a_n) of real numbers, there exists a limit a = lim(a_n), i.e, a_n converges
Convergent Subsequences Lemma Given a sequence (x_n) which is convergent , any subsequence (x_n1, x_n2, ... where n1<n2<n3...) is also convergent to the same limit
Cauchy A sequence (x_n) in a metric space (M,d) is said to be Cauchy if for every e>0, there is some N in N such that d(x_m, x_n) < e whenever m,n >= N
Convergence and Cauchy Fact A convergent sequence is cauchy. Idea: Let x = lim(x_n). Then d(x_m, x_n) <= d(x_m, x) + d(x_n, x) < e for large m, n in N
Bolzano-Weierstrass Theorem Suppose (a_n) is a bounded sequence of real numbers. Then there exists a convergent subsequence
Nested Interval Property Let I_1 = [a_1, b_1], I_2 = [a_2, b_2], ... be a sequence of closed finite intervals of real numbers. If the intervals are nested, i.e. I_1 contains I_2 contains I_3... then the intersection of all I_k is not empty. Hence, I_n = [a_n, b_n] contains [lim(a_n), lim(b_n)] for all n in N. Therefore the intersection of all I_k contains [a,b] which is non-empty
Cauchy Criterion Theorem A cauchy sequence in R is convergent
Show full summary Hide full summary


What were the Cause and Consequences of The Cuban Missile Crisis October 1962
AQA GCSE Product Design Questions
Bella Statham
yog thapa
Animal Farm CONTEXT
Lydia Richards2113
Derecho Aéreo
Adriana Forero
Flashcards de japonés - Unidad 3
Ignacio Regazzoli
Ley de la propiedad Intelectual en Venezuela
Ovidio Rodriguez
Yenith Diaz
alba yolanda gonzalez florez