Non-convergent Cauchy sequence in $\ell^1$ with respect to the $\ell^2$ norm

Question

Let $X = \ell^1$, the set of absolutely convergent real valued sequences and let $d_2(\mathbf{x},\mathbf{y}) = \left(\sum_{k=1}^\infty |x_k - y_k|^2\right)^{1/2}.$

This is the $2$-norm on the $1$ space.

Is $(X,d)$ complete? I don't think it is but I can't find an example of a non-convergent Cauchy sequence.

Answer 1

Consider the sequence $\mathbf x^{(n)}\in\ell^1\subset\ell^2$ defined by $$ x^{(n)}_k=\frac{1}{k}\textrm{ for } k\leq n,\quad x^{(n)}_k=0\textrm{ for } k > n. $$ Then $\mathbf x^{(n)}\overset{d_2}{\to} \mathbf x\in\ell^2\setminus\ell^1$, with $$ x^{(n)}_k=\frac{1}{k}\quad \forall k. $$ So $\ell^1$ is not $d_2$-complete.