Is it true that if $\sum_{1}^{\infty} |x_n|^3$ converges then $\sum_{1}^{\infty} \frac{|x_n|}{n}$ converges? Is it true that if $\sum_{1}^{\infty} |x_n|^3$ converges then $\sum_{1}^{\infty} \frac{|x_n|}{n}$ converges? I am trying to show whether $d(x_n,y_n) = \sum_{1}^{\infty} \frac{|x_n - y_n|}{n}$ is a metric over $l_3 = \{(x_n)_n: \sum_{1}^{\infty} |x_n|^3\ < \infty\}$. My attempt to show d is not a metric was to find a sequence such that $d(x,0) = \infty$ but I couldn't.

By Hölder's inequality we have $\sum\frac {|x_n|} n \leq  (\sum |x_n|^{3})^{1/3} (\sum \frac 1 {n^{3/2}})^{2/3} <\infty$
