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 stronger 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 suggestions approximately RContinuity Computing Limits of capabilities A evaluate of services non-stop services: ARead more...

summary: Sequences and Their Limits Computing the LimitsDefinition of the restrict homes of Limits Monotone Sequences The quantity e Cauchy Sequences restrict greater and restrict Inferior Computing the Limits-Part II genuine Numbers The Axioms of the Set R effects of the Completeness Axiom Bolzano-Weierstrass Theorem a few ideas approximately RContinuity Computing Limits of capabilities A assessment of capabilities non-stop services: a geometrical standpoint Limits of capabilities different Limits homes of continuing services The Continuity of straight forward capabilities Uniform Continuity homes of constant capabilities

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**Example text**

Then lim sup an = +∞ and lim inf an = 1. 9 that the set of accumulation points had the smallest one. Since we defined the limit inferior of a sequence as the smallest accumulation point, there had better be one. Of course, even if 0 was not an accumulation point, we could still call it the limit inferior, but the next theorem tells us that we do not need to worry about it. 10. For a bounded sequence {an }, there exists the largest and the smallest accumulation points. Cauchy used this result in Cours d’analyse, but he never proved it.

10. a1 = 0, a2 = 17 1 1 , an+2 = (1 + an+1 + a3n ). 11. Let c > 0 and a1 = , an+1 = (c + a2n ). Determine all c for which the sequence 2 2 converges. For such c find lim an . √ A 1 an + . Prove that {an } converges to A. 12. 5 Number e In this section we will consider a very important sequence. For that we will need some additional tools. The first one is an inequality. 1 (Bernoulli’s Inequality). If x > −1 and n ∈ N then (1 + x)n ≥ 1 + nx. Proof. We will use mathematical induction. When n = 1, both sides of the inequality are 1+x.

Prove that lim sup a1 + an+1 an n ≥ e. 9. Suppose that the terms of the sequence {an } satisfy the inequalities 0 ≤ an+m ≤ an + am . Prove that the sequence {an /n} converges. 6). In this chapter, our goal is to prove these theorems. When Cauchy did that in Cours d’analyse he took some properties of real numbers as self-evident. In the course of the 19th century it became clear that these needed to be proved as well, and for that it was necessary to make a precise definition of real numbers. This task was accomplished around 1872 by the independent efforts of Dedekind, Cantor, Heine, and M´eray.