Geometry

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Cauchy Sequences

         

Is the sequence {an}n=1∞\{a_n\}_{n=1}^{\infty} given by an=1na_n=\frac{1}{n} a Cauchy sequence?

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Is the sequence {an}n=1∞\{a_n\}_{n=1}^{\infty} given by an={1n if n is even1+1n if n is odda_n=\left\{ \begin{aligned} &\frac{1}{n}&&\text{ if }n\text{ is even}\\ &1+\frac{1}{n}&&\text{ if }n\text{ is odd}\end{aligned}\right. a Cauchy sequence?

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How many of the following statements are true?

I. If ana_n is a Cauchy sequence of rational numbers, then the limit of the ana_n is a rational number.

II. If ana_n is a Cauchy sequence of irrational numbers, then the limit of the ana_n is an irrational number.

III. If ana_n is a Cauchy sequence of real numbers, then the limit of the ana_n is a real number.

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If the sequence {an}n=1∞\{a_n\}_{n=1}^{\infty} is a Cauchy sequence, which of the following must also be Cauchy sequences? I.{1an}n=1∞II.{an2}n=1∞III.{sin⁡an}n=1∞\text{I.} \left\{\frac{1}{a_n}\right\}_{n=1}^{\infty}\quad \text{II.} \left\{a_n^2\right\}_{n=1}^{\infty}\quad\text{III.} \left\{\sin a_n\right\}_{n=1}^{\infty}

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How many of the following statements are true for a sequence of real numbers ana_n?

I. If lim⁡n→∞∣an−am∣=0\lim_{n\to\infty} |a_n-a_m|=0 for a single value of mm, then {an}n=1∞\{a_n\}_{n=1}^{\infty} is a Cauchy sequence.

II. If lim⁡n→∞∣an−am∣=0,\lim_{n\to\infty} |a_n-a_m|=0, for every mm, then {an}n=1∞\{a_n\}_{n=1}^{\infty} is a Cauchy sequence.

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