Take a number and map it to the number of letters in its name

Let f: N--> N map a number to the number-of-letters in its name in English (we will consider other languages later).

So for example 14 is fourteen so it maps to 7. Let's iterate

14 --> 7 --> 5 --> 4-->4-->4 ...

It is known (perhaps well-known) that, given any starting point, the sequence eventually is all 4's. 

I recently got an email asking if this was PROVEN.  I could not find a proof on the web so I did it myself. My writeup of a proof is here. (LANCE POINTED OUT A MISTAKE IN THIS DOCUMENT-WILL FIX LATER AND THEN REMOVE THIS COMMENT.) 

1) So now there IS a proof on the web. It may be hard to find. Does this problem have a well known name that I should add to this blog post so that it is easier to find?

2) My next thought was

For which functions f: N-->N is it the case that every sequence a, f(a), f(f(a)), ... is eventually constant. I leave that for you to ponder.

3) The asker of the question is of a more real-world bent and emailed me what happens in other languages:

Spanish has one number whose name is its own length: 5 is cinco. I leave it to the reader to show that in Spanish the sequence always becomes all 5's.

French seems to have no such numbers, so it cannot have this behavior.

Norwegian has three such numbers: 2 is to, 3 is tre, 4 is fire. So I suspect every such sequence either is constant-2, constant-3 or contant-4.

See this article here for more the numbers 1-10 in several languages.

OBLIGATORY ChatGpt note: Lance saw this post in our blog account and was curious what ChatGpt would do with it. For what he got see here. You can decide if a program can take over the blog anytime soon.