APMO 2001 Problem 1

For a positive integer $n$ let $S(n)$ be the sum of digits in the decimal representation of $n$ . Any positive integer obtained by removing several (at least one) digits from the right-hand end of the decimal representation of $n$ is called a stump of $n$ . Let $T(n)$ be the sum of all stumps of $n$ . Prove that $n=S(n)+9T(n)$ .