Convergence in $l^p$ space?

Question

We know that $\lim_{k \to \infty} a_{n}^{k} = b_n$. All $(a_{n}^{k}) \in l^p$ space. How to show that $(\lim_{k \to \infty} a_{1}^{k},\lim_{k \to \infty} a_{2}^{k},...,\lim_{k \to \infty} a_{n}^{k},...) = (b_1,b_2,...,b_n,...)$

Answer 1

It is not true in general that convergence in $\ell^p$ takes place: take $a_n^k=0$ if $n\neq k$ and $1$ otherwise.

However, if the remainders are uniformly convergent to $0$, that is, $$\lim_{N\to \infty}\sup_{k\geqslant 1}\sum_{n=N}^{+\infty}|a_n^k|^p=0,$$ then the convergence in $\ell^p$ holds by an approximation argument.