How can you prove that every natural number $M$ or $M+1$ can be written as $k + \operatorname{Sum}(k)$, where $\operatorname{Sum}(k)$ represents the sum of the digits of some number k.
Example: $$ 248 = 241 + \operatorname{Sum}(241) = 241 + 2 + 4 + 1$$
Hint: Each term in the sequence $k + \mbox{Sum}(k)$ either increases by 2, or decreases by some amount.
Why does that tell you that the image of $k + \mbox{Sum}(k)$ must include either $M$ or $M+1$?
Proof of Hint: If the last digit is not 9, then $k$ and $\mbox{Sum}(k)$ will both increase by 1, hence their sum increases by 2. If the last digit is 9, then $\mbox{Sum}(k)$ will decrease by at least 8, hence their sum decreases by sum amount.