The numbers 1000, 1001, ..., 2999 have been written on a board. Each time, one is allowed to erase two numbers, say \(a\) and \(b\), and replace by the number \(\frac {1}{2} \min(a, b)\).

After 1999 such operations, one obtains exactly one number \(c\) on the board. Prove that \(c < 1\).

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