By John Srdjan Petrovic

Sequences and Their Limits Computing the LimitsDefinition of the restrict homes of Limits Monotone Sequences The quantity e Cauchy Sequences restrict better and restrict Inferior Computing the Limits-Part II genuine Numbers The Axioms of the Set R results of the Completeness Axiom Bolzano-Weierstrass Theorem a few options approximately RContinuity Computing Limits of services A assessment of capabilities non-stop features: ARead more...

summary: Sequences and Their Limits Computing the LimitsDefinition of the restrict homes of Limits Monotone Sequences The quantity e Cauchy Sequences restrict more desirable 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 ideas approximately RContinuity Computing Limits of capabilities A assessment of services non-stop features: a geometrical perspective Limits of capabilities different Limits homes of continuing features The Continuity of undemanding capabilities Uniform Continuity homes of continuing capabilities

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Give an example to show that the inequality may be strict. Give an example to show that the assumption an , bn > 0 cannot be relaxed. 5. Let {an } be a sequence of positive numbers and let {bn } be a convergent sequence of positive numbers. Prove that lim sup(an bn ) = lim sup an lim bn . 6. Let {an } be an increasing sequence that has a bounded subsequence. Prove that the sequence {an } is convergent. 7. Let {an } be a sequence such that every subsequence {ank } contains a convergent subsequence {ankj } converging to L.

In fact, it is also bounded above by 9/5. Indeed, if this were not true, there would be a ∈ A such that a > 9/5. 24 > 2, contradicting the assumption that a2 < 2. Can we do better than 9/5? It is easy to verify that 8/5 is also an upper bound of A. Is there a least upper bound? The answer depends once again on the set in which we operate. If we allow real numbers, then √ the answer is in the affirmative, and we can even pinpoint the least upper bound: it is 2. ) Clearly, if we restrict ourselves to the set of rational numbers, then the answer is in the negative.

Prove that lim an = e. k=1 k(k + 1)(k + 1)! 8. Prove that the sequence an = 1 + + + · · · + − ln n is increasing and bounded 2 3 n above. Conclude that it is convergent. 7. 8. 5772. 9. 6. 10. A sequence {an } is a geometric sequence if there exists q > 0 such that an+1 /an = q, n ∈ N. A sequence {bn } is an arithmetic sequence if there exists d > 0 such that an+1 −an = d, n ∈ N. If {an } and {bn } are such sequences, and if a1 = b1 > 0, a2 = b2 > 0, prove that an > bn for n ≥ 3. Use this result to derive the Bernoulli’s Inequality.