Repeated Sum of Cubes of Digits of Multiple of 3

Theorem
Let $k \in \Z_{>0}$ be a positive integer.

Let $f: \Z_{>0} \to \Z_{>0}$ be the mapping defined as:


 * $\forall m \in \Z_{>0}: \map f m = $ the sum of the cubes of the digits of $n$.

Let $n_0 \in \Z_{>0}$ be a positive integer which is a multiple of $3$.

Consider the sequence:
 * $s_n = \begin{cases} n_0 & : n = 0 \\

\map f {s_{n - 1} } & : n > 0 \end{cases}$

Let $n \in \Z_{>0}$ be a positive integer which is a multiple of $3$.

Then:
 * $\exists r \in \N_{>0}: s_r = 153$

That is, by performing $f$ repeatedly on a multiple of $3$ eventually results in the pluperfect digital invariant $153$.