EXTRA CREDIT, Fall Semester 2001

There is an offer of 500 POINTS out there for anybody who can prove that the sequence from last week's Internet Special always gets caught up in 1 - 2 - 1 - 2 - 1 - 2 ..., no matter what the starting value (a natural number, i.e., a positive integer) is. The deadline for this problem is Friday, December 14, 2001, 5:30 pm.