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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