 Geometry

# Cauchy Sequences

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

Is the sequence $\{a_n\}_{n=1}^{\infty}$ given by a_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?

How many of the following statements are true?

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

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

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

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

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

I. If $\lim_{n\to\infty} |a_n-a_m|=0$ for a single value of $m$, then $\{a_n\}_{n=1}^{\infty}$ is a Cauchy sequence.

II. If $\lim_{n\to\infty} |a_n-a_m|=0,$ for every $m$, then $\{a_n\}_{n=1}^{\infty}$ is a Cauchy sequence.

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