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This paper gives a simple proof of the Wirsching-Goodwin representation ofintegers connected to 1 in the $3x+1$ problem (see \cite{Wirsching} and\cite{Goodwin}). This representation permits to compute all the ascendingCollatz 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.

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