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.

If you don't know about Holder inequality, you can still pull it off with a low key AM-GM inequality. Note that $\dfrac{|x_n|}{n}=\dfrac{1}{\sqrt{n}}\cdot \dfrac{1}{\sqrt{n}}\cdot |x_n|\le \dfrac{1}{3}\cdot \left(2\dfrac{1}{n^p}+|x_n|^3\right)$, with $p = \dfrac{3}{2}$. Thus the series in question converges by comparison test.
