Carrying digit can only be 1. Why?

Question

In the book "Algebra" by Gelfand, I have read a solution to a problem which assumes something that does not seem evident to me. Let's see:

Problem 2. In the addition example:

  A A A +
  B B B =
---------
A A A C

all A's denote some digit, all B’s denote another digit and C denotes a third digit. What are these digits?

Solution. First of all A denotes 1 because no other digit can appear as a carry in the thousands position of the result. [...]

How do the authors know that?

Answer 1

Since $AAA \le 999$ and $BBB \le 999$, necessarily their sum is smaller or equal to $999+999 = 1998$. So the thousands digit must be at most $1$.