How to prove convergence of a sequence of binary numbers

Question

I have a boolean expression with 4 inputs and 1 output, that when iterated onto itself(output->input(s)), the function converges to 1. How do I go about proving the convergence of a sequence of binary numbers?

Ive been playing around with the idea of learning more about statistical analysis to find out if there are hints there, since I can generate a very big dataset of binary numbers by that expression.

Or should I learn more about binary trees? And try to solve it that way?

Just to give a simple example of what I mean: a set of integers: {10,9,8,7,6,5,4,3,2,1}. in binary:

1010
1001
1000
0111
0110
0101
0100
0011
0010
0001

at final step it produce 1. So by the dataset it is easy to see that. But how do one go around proving that by a formula like a boolean expression?

I want to be able to understand proof-techniques for integers like binary 2-state boolean expressions rather than plain decimal numbers

Thanks for your time

Answer 1

The "function" you use is $a_{n+1}= a_n- 1$. Given any positive initial number that gets every number less that the initial number so certainly 1.