Smallest Multiple of 9 with all Digits Even/Proof 2

Proof
Let $n$ be the smallest integer multiple of $9$ all of whose digits are even.

From Divisibility by 9, the digits of $n$ must add to an integer multiple of $9$.

But from Sum of Even Integers is Even, the digits of $n$ must add to an even integer multiple of $9$: $18, 36, 54$, etc.

There is only $1$ integer multiple of $9$ with $2$ digits whose digits add to an even integer multiple of $9$, and that is $99$.

Thus we have to look at $3$-digit integers.

The following sets of $3$ even digits add to $18$:
 * $\set {2, 8, 8}$
 * $\set {4, 6, 8}$
 * $\set {6, 6, 6}$

and that seems to be about it.

There are no sets of $3$ even digits which add up to $36$ or higher.

The result follows by inspection.