TITLE: Proving an interesting feature of any $1000$ different numbers chosen from $\{1, 2, \dots,1997\}$
QUESTION [5 upvotes]: Assume you choose $1000$ different numbers from the group $\{1, 2,
\dots,1997\}$.
Prove that within the $1000$ chosen numbers, there is a couple which
  sum is $1998$.

I defined:

pigeonholes: possible sums.
pigeons: the $1000$ different numbers.

Is this definition good or there is something better?

REPLY [2 votes]: I would consider the sets $A_1 = \{1,1997\}, A_2 =\{2,1996\} \ldots A_{998} = \{998, 1000\}, A_{999}=\{999\}$. Note that they contain all the numbers ${1,\ldots , 1997}$.
Now, we want to choose $1000$ numbers: it means that you take at least two elements from the same set and hence their sum is indeed $1998$.