Consider a series of numbers, defined as follows: Starting with any natural number, each member is a sum of the squares of the previous member`s digits.

Prove : The series always reaches either a stuck-on-one sequence: 1,1,1… or a closed loop of the following 8 numbers: 145,42,20,4,16,37,58,89, ...

"Jer understood that only 99 (and not 999 ) numbers have to be examined (need I explain WHY?) "

I have realized that 999 was overkill, but I don't see that 99 would suffice. Even 3*81=243 allows that a 3-digit number could produce another 3-digit number and there might be a cycle wholly within the 100-243 range. Certainly a proof would have to spell out what may seem obvious to you.