Repdigit Number consisting of Instances of 9 is Kaprekar

Theorem
A repdigit number that consists entirely of the digit $9$ is a Kaprekar number.

Proof
We note as examples:

Now we consider:

So $\paren {10^{n + 1} - 1}^2$ can be split as:


 * $\paren {10^{n + 1} - 2} + 1 = 10^{n + 1} - 1 = \displaystyle \sum_{k \mathop = 0}^n 9 \times 10^k$

Thus $\displaystyle \sum_{k \mathop = 0}^n 9 \times 10^k$ is a Kaprekar number.