Geometry
# Topology

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

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.