What are some examples of functions that given a positive integer x converge to 1 using only natural operations (add, sub, div, mul).
An example would be the collatz conjecture, although it requires the use of modular arithmetic so it's not the best example.
An intuitive example is the function $f(x)=1$.
Of course your assertion that the Collatz map on $\mathbb{Z}$ will converge all positive integers to $1$ has never been proven, so it does not serve as a good example.