Here comes another trivial proof (the base must be at least 2 for the obvious reason):
Suppose x, y, z are single-digit numbers in base b ≥ 2. By the definition, , , and . Thus, the sum has the property .
In this case, the sum can be expressed in single digit.
Let and . Then and . Because b ≥ 2, . It follows that and .
Using the property of the ceiling function:
It follows that or that . Now multiply −1 to both sides to get and . Since are integers, . Thus, and .
Given the above, the sum x + y + z can be expressed as where and . Hence can be expressed in two digits.
Thus in both cases, the sum of three numbers x, y, z can be at most two digits long. Q.E.D.