Theorems and Definitions
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| show | If a sequence is monotone and bounded, then it converges.
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| show | Every nonempty set of real numbers that is bounded above has a least upper bound.
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| show | 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.
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| Archimedean Property | show 🗑
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| show | For every two real numbers a and b with a<b, there exists a rational number r satisfying a<r<b.
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| show | Given any set A, there does not exist a function f: A-->P(A) that is onto.
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| Algebraic Limit Theorem | show 🗑
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| Order Limit Theorem | show 🗑
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| Bolzano-Weierstrass Theorem | show 🗑
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| show | 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.
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| show | The set A is bounded below if there exists a lower bound l∈R satisfying l≤a for every a∈A.
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| Supremum | show 🗑
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| Maximum | show 🗑
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| Same cardinality | show 🗑
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| Countable/ Uncountable | show 🗑
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| show | A sequence is a function whose domain is N.
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| show | 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.
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| show | 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.
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| Diverge | show 🗑
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| show | A sequence (x_n) is bounded if there exists a number M>0 such that |x_n|≤M for all n∈N.
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| show | 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.
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| Monotone | show 🗑
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| show | 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.
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| show | 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.
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| Divergence Criterion | show 🗑
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