The question is as follows
Prove $\frac{1}{n}\in\ell_p$ for any $1<p<\infty$ is not product of two elements
My Solution.
Suppose there exist $(y_n)_{n=1}^{\infty}$ and $(z_n)_{n=1}^{\infty}$ in $\ell_p$ such that $\frac{1}{n}=(y_nz_n)_{n=1}^{\infty}$, so we have $$z_n=\frac{1}{ny_n}, \quad\sum_{n=1}^{\infty}|y_n|^p<\infty$$ and so $$\sum_{n=1}^{\infty}|z_n|^p=\sum_{n=1}^{\infty}|\frac{1}{ny_n}|^p$$
I want to get a contradiction How do I show that?
Can you please help me?