Here is something that I was certain would have an easy-to-prove positive answer; but it turns out that there is an exceedingly simple counterexample.

Let be the first infinite ordinal; its successor is .We consider a map , or put differently, a -Matrix of real numbers.

We assume that has the following properties:

(1) For all we have , or more informally, we have “convergence to the right”.

(2) For all we have , or more informally, we have “convergence to the bottom”.

(3) , or more informally, the right-hand entries of converge to the bottom right element .

**Question:** Does this imply that , that is, do the bottom entries of also converge to ?

**Answer:** There is, surprisingly (to me, at least) an easy example showing that the answer is **No**. Let be defined by if and otherwise. It is easy to verify that (1), (2), (3) above are satisfied. Note that the right-hand entries are all 0, and they trivially converge to ; but we have for all therefore .

For the kind of two-dimensional convergence we are looking for we need some form of “simultaneous” convergence (conceptually related to uniform convergence), which I might address in a later post.