Cauchy sequence- series (TIFR-2013 problem 11)

Question:

True/False:

Consider the sequences $$x_n=\sum_{1}^{n} \frac{1}{j}$$ and $$y_n=\sum_{1}^{n} \frac{1}{j^2}$$. Then $$\left\{x_n\right\}$$ is Cauchy but $$\left\{y_n \right\}$$ is not.

Discussion: We are given sequence of partial sums of a very well known type of series.  $$\left\{x_n\right\}$$ is a divergent sequence and $$\left\{y_n\right\}$$ is convergent. Also, $$\mathbb{R}$$ is complete. So every Cauchy sequence is convergent and any convergent sequence (as always happens in metric spaces) is Cauchy.

The true statement would be $$\left\{x_n\right\}$$ is not Cauchy and $$\left\{y_n\right\}$$ is Cauchy sequence.