Find a sequence in $\ell^p(\mathbb R)$ such that each component converges to zero but the sequence itself does not converge in $\ell^p(\mathbb R)$ Find a sequence in $\ell^p(\mathbb R)$, $1\leq p\leq\infty$ such that each component converges to zero but the sequence itself does not converge in $\ell^p(\mathbb R)$, where $\ell^p$ consists of real sequences $x_k$ with $(\sum_{k=1}^{\infty}\lvert x_k\rvert^p)<\infty$.
I'm struggling to understand exactly what this question is asking. I think it's asking me to find a sequence of sequences, $x_k^{(n)}$ that each converge to zero as $k\to\infty$ but has no limit in $\ell^p(\mathbb R)$ as $n\to\infty$. I'm also not sure whether it wants me to find a sequence so that this holds for all $p$.
I tried the sequence of sequences
$x^1=(1,0,0,0,...)\\x^2=(1,2,0,0,0,...)\\x^3=(1,2,3,0,0,0,...)$
where each $x^{(n)}\in\ell^p(\mathbb R)$ for each $n$ and all $1\leq p\leq \infty$, and each sequence tends to zero, but as $n\to\infty$ the limit is not in $\ell^p(\mathbb R)$. Have I interpreted the question correctly and if I have does this make any sense?
 A: The question is asking for a sequence
$$
  x^{(1)}, x^{(2)},x^{(3)},\ldots ,
  $$
where each $x^{(n)} = \big (x^{(n)}_k\big )_{k\in {\mathbb N}}$ is an element of $\ell ^p$, and such that

*

*$\displaystyle\lim_{n\to \infty }x^{(n)}$ does not exist in $\ell ^p$.


*For each fixed $k$,  one has that $\displaystyle\lim_{n\to \infty }x^{(n)}_k = 0$.
The second condition means that each column, below,  converges to zero.
$$
\matrix{
x^{(1)} & = & \big(x^{(1)}_1,x^{(1)}_2,x^{(1)}_3,x^{(1)}_4,\ldots \big) \cr
x^{(2)} & = & \big(x^{(2)}_1,x^{(2)}_2,x^{(2)}_3,x^{(2)}_4,\ldots \big) \cr
x^{(3)} & = & \big(x^{(3)}_1,x^{(3)}_2,x^{(3)}_3,x^{(3)}_4,\ldots \big) \cr
x^{(4)} & = & \big(x^{(4)}_1,x^{(4)}_2,x^{(4)}_3,x^{(4)}_4,\ldots \big) \cr
}
$$
A: The classic "mass running to $\infty$" example:
$$x^{(1)} = (1, 0, 0, 0, \ldots)$$
$$x^{(2)} = (0, 1, 0, 0, \ldots)$$
$$x^{(3)} = (0, 0, 1, 0, \ldots)$$
$$x^{(4)} = (0, 0, 0, 1, \ldots)$$
and so on...
Every fixed component is eventually constant $0$, so each component converges, but the sequence does not converge in $\ell^p$ for any $p \in [1, \infty]$.
