Convergence of a Cauchy sequence of matrices I have a Cauchy sequence of matrices $C_i \in R^{p \times q}$, i.e. $\lim_{n\rightarrow \infty} \| C_{n+1} - C_{n} \| = 0$ for any norm (I just need the property that $\|C_1-C_2\|>\delta \Rightarrow C_1 \neq C_2$). I also know $\|C\|_{1,1} \leq t$ for a fixed $t$ (where $\|C\|_{1,1} = \sum_{i,j} |c_{ij}|$).
Can I conclude that the sequence $C_i$ also converges? (i.e., is the corresponding metric space complete?)

If a sequence in this space is Cauchy, then it converges.  That is, $\Bbb R^{p \times q}$ under $\|\cdot\|_{1,1}$ (which is isometric to $\Bbb R^{pq}$ under $\|\cdot\|_1$) is indeed a complete metric space.
In general: the finite Cartesian product of complete metric spaces will be complete (this is not true, however, for arbitrary products).
That being said, the condition you provided is insufficient to guarantee that the sequence is Cauchy. As a counterexample, consider the sequence in $\Bbb R^{2 \times 1}$ given by
$C_n = (\cos \theta_n,\sin\theta_n)$ where
$$
\theta_n = \sum_{i=1}^n \frac 1i
$$
Noting that $\|C_n\| < \sqrt{2}$ for each $n$.
