Decimal number presentation

[elim]

(1) For $n = \sum_0^{k-1} a_j 10^j, \quad 10 > a_j \in \mathbb{N}, \quad j=\overline{0,k-1}$ define $S(n) = \sum_0^{k-1} a_j$
$\quad\quad$ Prove that $\forall m \in \mathbb{N} \quad \exists n \in \mathbb{N}:\; n+S(n)\in \{m,m+1\}$