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By John Srdjan Petrovic

Sequences and Their Limits Computing the LimitsDefinition of the restrict houses of Limits Monotone Sequences The quantity e Cauchy Sequences restrict more suitable and restrict Inferior Computing the Limits-Part II actual Numbers The Axioms of the Set R effects of the Completeness Axiom Bolzano-Weierstrass Theorem a few techniques approximately RContinuity Computing Limits of capabilities A assessment of features non-stop capabilities: A Read more...

summary: Sequences and Their Limits Computing the LimitsDefinition of the restrict houses of Limits Monotone Sequences The quantity e Cauchy Sequences restrict greater and restrict Inferior Computing the Limits-Part II actual Numbers The Axioms of the Set R outcomes of the Completeness Axiom Bolzano-Weierstrass Theorem a few techniques approximately RContinuity Computing Limits of capabilities A overview of capabilities non-stop features: a geometrical standpoint Limits of features different Limits homes of constant features The Continuity of common services Uniform Continuity houses of continuing features

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That converges to 0. Therefore, the set of accumulation points includes 0 as well. It is of interest to learn about the largest and the smallest accumulation points of a sequence. We call them the limit superior and the limit inferior of the sequence {an }, and we write lim sup an (or lim an ) and lim inf an (or lim an ). It is convenient to include unbounded sequences in the discussion, in which case lim sup an and lim inf an may turn out to be infinite. For example, if the sequence {an } is 1, 2, 1, 3, 1, 4, 1, 5, .

First we notice that 0 < 2 − a2 < 2 and 2a + 1 > 1, so 0 < ε < 1 and, consequently, ε2 < ε. Therefore, (a + ε)2 = a2 + 2aε + ε2 < a2 + 2aε + ε = a2 + (2a + 1)ε = a2 + < 2 − a2 2 + a2 = 2 2 2+2 = 2. 2 Thus, a + ε ∈ A. Since ε > 0, we see that a + ε > a, contradicting the assumption that a is an upper bound of A. Now we turn our attention to the other possibility: a2 > 2. We will show that a is “too big,” meaning that we can find an upper bound of A that is smaller than a. We claim that one such number is a − ε, where a2 − 2 ε= .

Jacob Bernoulli was one of the many prominent mathematicians in the Bernoulli family. Following his father’s wish, he studied theology and, contrary to the desires of his parents, mathematics and astronomy. He became familiar with calculus through a correspondence with Leibniz, and he made significant contributions (separable differential equations), as well as in probability (Bernoulli trials). He founded a school for mathematics and the sciences at the University of Basel and worked there as a professor of mathematics for the rest of his life.

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