Presumably, one would have to first define a standard naming convention for numbers. This is a completely non-trivial task. For instance, is 1000 spelled out "one thousand" or "one times ten to the third"? The second "spelling" might appear to be cheating, but realize that there are infinite numbers, and we only finitely many named units (thousands, millions, billions, trillions, etc.,.). In fact, even if we resort to a naming convention based on scientific notation, we would still eventually run out of ways to name the exponent. In fact, the only sensible naming convention that I can think of is naming the digits placewise either from left to right or right to left. If we go off of this naming convention, this makes things a lot easier.

It seems that under this naming convention, the number of letters in the spelling representation of numbers tends to grow slower than the numbers themselves. Let #x represent the number of letters in the spelling representation of the number x. I hypothesize that there exists some number n such that for all m > n, m > #m. If this is true, we would simply have to check that all numbers less than n+1 and make sure that they converge to 4, and this would be sufficient to show that all numbers converge to 4.