A simple proof of the Wirsching-Goodwin representation of integers connected to 1 in the 3x+1 problem

This paper gives a simple proof of the Wirsching-Goodwin representation of integers connected to 1 in the $3x+1$ problem (see \cite{Wirsching} and \cite{Goodwin}). This representation permits to compute all the ascending Collatz sequences $(f^{(i)}(n),\: i=1,b-1)$ with a last value $f^{(b)}(n)=1.$ Other periodic sequences connected to $1$ are also identified.