I know the question has been asked here, but I do not understand the solution (Are $\ell_p$ spaces complete under the $q$-norm?)
I came up with my own solution and was wondering if it is correct.
Denote $l^p_q$ as $l^p$ with $q-$norm. Now if it is closed, then the identity map $l^p_p \to l^p_q$ is a continuous bijection (easy to show), and so by open mapping theorem the inverse mapping is bounded. Therefore $\exists M$ s.t $\|x\|_q\leq M\|x\|_p$. Now consider $x_n=(2^n,0,0....)$ then $\|x_n\|_q=2^{nq}$ and $\|x_n\|_p=2^{np}$ thus $2^{n(q-p)}\leq M$ for all $n$ which is a contradiction.
Is this correct? Is there an even simpler solution?
There are several mistakes. Continuity of the inverse gives $\|x\|_p \leq M\|x\|_q$ (not $\|x\|_q \leq M\|x\|_p$). Besides your computation of the norms is wrong. In your example $\|x\|_p=\|x\|_q=2^{n}$.
Consider the sequences $(1, 2^{-1/p},3^{-1/p},...., N^{-1/p},0,0,...)$ to arrive at a contradiction to the inequality $\|x\|_p \leq M\|x\|_q$