Embedding $\mathcal{l}^1 \subset \mathcal{l}^p$ continuous

Question

Is the embedding of $\mathcal{l}^1$ in $\mathcal{l}^p$ for any $p\geq 1$ continuous?

Answer 1

Yes. if $x_n\in\ell^1$ converges to $0\in\ell^1$, then for large enough $k$, $|(x_n)_j|<1$ for all $j\in\Bbb N$ and $n>k$. But then $$ \sum_j |(x_n)_j^p|\leq \sum_j |(x_n)_j| \to 0. $$