There's some sleight of hand here. Not all the digits are exactly right. Look how it skips from 997 to 999:

http://www.futilitycloset.com/2012/01/08/math-notes-76/

Here's the math. Suppose you want a unit fraction 1/n with decimals that cycle through the 4-digit sequence abcd. Multiply by 10^4 to shift abcd into integer position, leaving repeating copies after the decimal point:

    10^4/n = abcd + 1/n

Solving for n gives n = (10^4 - 1) / abcd.

More generally, if you want to get a cycle equal to the d digits of an integer k, you want n = (10^d - 1) / k.

However, this only gives a true unit fraction when k divides 10^d - 1, so that n is an integer. Otherwise you are forced to truncate n and getting an approximate version of the cycle.

That's exactly what happened here: 10^d - 1 is not divisible by the integer 001002...998999.

Here's a small Python program that will generate the unit fraction given the number of digits to cycle through:

    import sys

    n = int(sys.argv[1])
    s = ''.join(("%0" + str(n) + "d") % (i,) for i in range(10**n))
    print "1/%d" % ((10**len(s) - 1) / int(s),)

Usage:

    $ python magic.py 3
    1/998001
    $ python magic.py 4
    1/99980001

This has just the right flavor for a Project Euler problem.