This is a complement of my previous question about the sequence spaces (I'm afraid, there will be a third part).

Let $\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty] $ satisfy the following properties:

$\rho(\lambda u)=\lambda\rho( u)$, for every $u\in [0,+\infty)^{\mathbb{N}}$ and $\lambda\ge 0$;

If $\rho( u)=0$, then $u=0_{\mathbb{R}^{\mathbb{N}}}$;

$\rho(u+v)\le \rho( u)+\rho( v)$, for every $u,v\in [0,+\infty)^{\mathbb{N}}$.

$\rho(\{u_n\}_{n\in\mathbb{N}})\le \rho(\{v_n\}_{n\in\mathbb{N}})$, once $0\le u_n\le v_n$, for all $n$, and the set $\{\frac{u_n}{v_n}, n\in \mathbb{N}\}$ is finite (we count $\frac{0}{0}=0$).

Define $\|\cdot\|:\mathbb{R}^{\mathbb{N}} \to[0,+\infty] $ by $\left\|\{u_n\}_{n\in\mathbb{N}}\right\|=\rho(\{|u_n|\}_{n\in\mathbb{N}})$.

Does it follow that $E=\{u\in \mathbb{R}^{\mathbb{N}}, \|u\|<+\infty\}$ is a linear space and $\|\cdot\|$ is a norm on $E$?