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# Real Analysis Ch.1-2

### Theorems and Definitions

Monotone Convergence Theorem If a sequence is monotone and bounded, then it converges.
Axiom of Completeness Every nonempty set of real numbers that is bounded above has a least upper bound.
Approximation Property of the Supremum Assume s∈R is an upper bound for a set A⊂R. Then, s=supA iff, for every choice of ϵ>0, there exists an a∈A satisfying s-ϵ<a.
Archimedean Property 1) Given any number x∈R, there exists an n∈N satisfying n>x. 2)Given any real number y>0, there exists an n∈N satisfying 1/n<y.
Density of Q in R For every two real numbers a and b with a<b, there exists a rational number r satisfying a<r<b.
Cantor's Theorem Given any set A, there does not exist a function f: A-->P(A) that is onto.
Algebraic Limit Theorem Let lim(a_n)=a and lim(b_n)=b. Then: 1) lim(ca_n)= ca, for all real numbers c 2) lim(a_n+b_n)= a+b 3) lim(a_n*b_n)= ab 4) lim(a_n/b_n)= a/b (when b does not equal 0)
Order Limit Theorem Assume lim(a_n)=a and lim(b_n)=b. 1) If a_n≥0 for all n∈N, then a≥0. 2) If a_n≤b_n for all n∈N, then a≤b. 3) If there exists c∈R for which c≤b_n for all n∈N, then c≤b. Also, if a_n≤c for all n∈N, then a≤c.
Bolzano-Weierstrass Theorem Every bounded sequence contains a convergent subsequence.
Bounded Above/ Upper Bound A set A∈R is bounded above if there exists a number b∈R such that a≤b for all a∈A. b is an upper bound for a.
Bounded Below/ Lower Bound The set A is bounded below if there exists a lower bound l∈R satisfying l≤a for every a∈A.
Supremum A real number s is the supremum (least upper bound) for a set A∈R if: 1) s is an upper bound for A; 2) if b is any upper bound for A, then s≤b.
Maximum A real number a_0 is a maximum for a set A if a_0 is an element of A and a_0≥a for all a∈A.
Same cardinality Two sets A and B have the same cardinality if there exists a function f:A-->B that is one to one and onto.
Countable/ Uncountable A set A is countable if A has the same cardinality as N. An infinite set that is not countable is uncountable.
Sequence A sequence is a function whose domain is N.
Convergence of a Sequence A sequence (a_n) converges to a real number a if, for every positive number ϵ, there exists a natural number N such that whenever n≥N it follows that |a_n-a|<ϵ.
Convergence of a Sequence (Topological Version) A sequence (a_n) converges to a if, given any ϵ-neighborhood V_ϵ(a) of a, there exists a point in the sequence after which all of the terms are in that ϵ-neighborhood.
Diverge A sequence that does not converge is said to diverge.
Bounded A sequence (x_n) is bounded if there exists a number M>0 such that |x_n|≤M for all n∈N.
Increasing/ Decreasing A sequence (a_n) is increasing if a_n≤a_n+1 for all n∈N. The sequence is decreasing if a_n≥a_n+1 for all n∈N.
Monotone A sequence is monotone if it is either increasing or decreasing.
Sequence of Partial Sums The sequence of partial sums is s_m=b_1+b_2+b_3+...+b_m. The infinite series converges to B if the sequence of partial sums s_m converges to B.
Subsequence Let (a_n) be a sequence of real numbers, and let n1<n2<n3<... be an increasing sequence of natural numbers. Then the sequence a_n1, a_n2, a_n3,... is called a subsequence of (a_n) and is denoted by (a_nj), where j∈N indexes the sequence.
Divergence Criterion If two subsequences converge to different limits, then the original sequence diverges.
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