Lemma Statement: \[ s(R_m \cdot k) = 9m,\text{where } R_m = \underbrace{99\ldots9}_{m \text{ times}} \text{ and } k \leq R_m + 1. \] Let \( M = \{ R_m, 2 \cdot R_m, \ldots, n \cdot R_m \} \), for some \( m \), I affirm that the set $M$ satisfies the requirements of the problem. Consider any subset \( S \subseteq M \). The sum of the elements of \( S \) is of the form \[ S_{S} = R_m \cdot R, \]where \( R \) is the sum of coefficients corresponding to the elements of \( T \). Clearly, \( R \) satisfies \( 1 \leq R \leq \frac{n(n+1)}{2} \). Select \( m \) such that \[ \frac{n(n+1)}{2} < R_m. \]This is always possible because \( R_m \) grows exponentially with \( m \), while \( \frac{n(n+1)}{2} \) is a fixed value for a given \( n \), then the $tacloyo$ of every subset of $M$ is $9m$, we are done.