Closed ball in $l_p$ is also closed in $l_q$

Question

I am trying to figure out if a closed unit ball in $l_p$ is also closed in $l_q$ for $1 \le p < q < \infty $. It looks easy at a first glance, but I got stuck pretty soon. I supposed there's a converging sequence (with respect to $||.||_{q}$), but we don't know whether it is Cauchy with respect to $||.||_{p}$, since $||x||_{p} \ge ||x||_{q}$, and there're no other nice inequalities between them. Any hints please?

Answer 1

If $(x_k)$ is a sequence of vectors in $\ell_q$, each with $p$-norm at most one, that converges to $x$ in $\ell_q$, then $(x_k)$ converges to $x$ coordinatewise.

For each $k$ and $n$, $\sum\limits_{i=1}^n |x_k(i)|^p\le 1$. Fix $n$ and let $k\rightarrow\infty$ to deduce that $\sum\limits_{i=1}^n |x(i)|^p\le1$ for each $n$.

Now let $n\rightarrow\infty$.