Topology
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.