Consider the sequence $(x^k)$ in $c_{00}$ defined by
$\begin{align} x^1 &= (1,0,0,0,\ldots) \\ x^2 &= (1,1,0,0,0,\ldots) \\ x^3 &= (1,1,1,0,0,0,\ldots), \end{align}$ and so on.
With respect to the sup norm, the sequence does NOT converge to the "obvious guess" $x = (1,1,1,1,1,1,\ldots)$ in the space $c_{00}$, because $(1,1,1,1,1,1,\ldots)$ is not in $c_{00}$. Neither does it converge to $x$ in a larger space with the sup norm. Neither does is converge to a larger space with a different norm that I can think of (e.g. the quadratic mean norm).
This is seemingly horrible. What is going on here!?! It SHOULD converge in SOME SPACE! Come on!
Is there a norm $\|\cdot\|$ so that $x^k$ converges to $x$ in a larger space?
My only guess is the "inf norm", but I'm not even sure that is actually a norm. (Actually, I'm quite certain it isn't because we would then have $\|(1,0,0,0,\ldots)\|_\inf = \|(2,0,0,0,\ldots)\|_\inf = 0$, contradicting one of the norm axioms). Actually, this is a stupid guess, but I'm out of ideas...
Also, intuitively, why do we not have $x^k \to x$ in $c_{00}$ w.r.t. all the standard norms mentioned above? Remember, humans introduced these norms, and why have we introduced all these norms if it doesn't have some of the most "obvious" convergence properties (like the one I am after). Is it something to do with the fact that $c_{00}$ is incomplete in all the standard norms? Or is it just because of the nature of the space $c_{00}$? Is the space $c_{00}$ not a "nice" one? By standard norms I mean the sup norm $\|\cdot\|_{\infty}$ and the norms $\|\cdot\|_p$ where p is an integer.
EDIT:
I think I have found an IDEA (one that I am not used to). Maybe we could think of limit inferior or limit superior in some way to come up with a suitable norm.