Show that $\ell^1$ is non complete under $\ell^2$ norm I'm trying to show that the metric space $ \ell^1$ is not complete under $\left(\sum_{k=1}^\infty |x_k - y_k|^2\right)^{1/2}.$
So I'm trying find a non convergent cauchy sequence in $\ell^1$ w.r.t  $\left(\sum_{k=1}^\infty |x_k - y_k|^2\right)^{1/2}.$ 
An example was given here Non-convergent Cauchy sequence in $\ell^1$ with respect to the $\ell^2$ norm , but I dont seem to understand why it is not convergent in the $\ell ^1 space $. 
My understanding is the sequence goes something like 
$(1 , \frac{1}{2} , \frac{1}{3}, ...... \frac{1}{k} , 0 , 0 , 0 , 0 , . . .  )$ 
The $d_{2}(x^n,x^m)$ ---> 0 for some large m,n where m > n so the sequence is obviously Cauchy. 
However the sequence seems to converge to $0$ $\in \ell^1$. So I dont understand why this sequence is non convergent in $\ell^1$
Is this sequence correct? If not what is another example of a non convergent cauchy sequence in $\ell ^1$ 
If someone could clarify that would be appreciated 

The example works. Indeed the limit is the harmonic sequence which does famously not belong to $\ell^1$.
For the sake of conventions, $\ell^p$ is used for sequences and $L^p$ for functions.
