Consider the space $M_\infty(\mathbb{C})$ of infinite matrices with only finitely many nonzero entries. This is a pre-C$^*$-algebra, and we can consider its unitization $M_\infty^1(\mathbb{C})$ (the unit will be an infinite matrix with ones on the whole diagonal). From what I understand, this space is not complete (its completion is the C$^*$-algebra $\mathcal{K}$ of compact operators on a separable infinite-dimensional Hilbert space).
Why is $M_\infty^1(\mathbb{C})$ not complete? What is an example of a Cauchy sequence in this space that does not converge?
Consider the matrices $x_n$ given by $$x_n=1\oplus\frac12\oplus\cdots\oplus\frac1{2^{n-1}}\in M_n(C)$$ Then the sequence $(x_n)_{n\in\mathbb N}$ is Cauchy in $M_\infty(\mathbb C)$, with $$\|x_n-x_m\|=\frac{1}{2^{\min\{m,n\}}},$$ but $(x_n)_{n\in\mathbb N}$ does not converge in $M_\infty(\mathbb C)$ (or its unitization), because any such limit would have $(n,n)$-entry given by $\frac{1}{2^{n-1}}$, hence has infinitely many nonzero entries.