Assume that there is a way to to this without taking any numbers with difference $4$. Note that we can take at most $4$ numbers from every group of $8$ consecutive numbers, because of $\pmod{8}$. This means that we can take at most half of the first $1992$ numbers, which is $996$. We can take at most $4$ numbers from $\{1993, 1994, 1995, 1996, 1997\}$, for a total of at most $1000$ numbers, which is too small. This is a contradiction, so we are done.