Math 117 Midterm 1 Terms

esnyder1994
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esnyder1994
Created by esnyder1994 over 5 years ago
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First half of terms covered by Math 117: Introduction to Analysis at UCSB
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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
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