Show whether the sequence is a Cauchy Sequence

Question

How can I show that $x_n= \sin(\frac{n\pi}{3})+\frac{1}{n}$ is a Cauchy Sequence and if it is not, finding a subsequence which is a Cauchy?

Answer 1

Note that \begin{equation} \label{eq:aqui-le-mostramos-como-hacerle-la-llave-grande} \sin(\frac{n\pi}{3}) = \left\{ \begin{array}{ll} (-1)^k\frac{\sqrt3}{2} & \mathrm{if\ } n=3k+1\;, n=3k+2 \\ 0 & \mathrm{if\ } n=3k\; \end{array} \right. \end{equation} So there are three subsequences which have different limits, so the original sequence cannot be convergent, hence it cannot be Cauchy. For example the subsequence $x_{3k}=\frac{1}{3k}$ is Cauchy because it converges to zero