Algebra
# 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.

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