Let's say the weight of a number is the sum of its digits. What is the smallest number that weighs 2000?
Observe that the numbers 189, 8307, and 99 all have a weight of 18, and that 99 is the smallest number with a weight of 18. To increase the weight of a number and minimize the number, the number should have as many digits 9 as possible. On the other hand, we can say that the digit 0 is eliminated from the digits to be considered, because it increases the number without increasing its weight.

We have that $2000=9 \times 222+2$. Therefore, the number we are looking for must have 222 digits 9 and one digit 2, or two digits 1. We eliminate the case of numbers with two digits 1 because they have 224 digits, making them larger than the numbers that have the digit 2 and have 223 digits. Thus, the number we are looking for has a 2 followed by 222 digits 9: the number is 299 . . 999.