For all 
,
there exists a 
such that the 
th
term of the Goodstein sequence 
. In other words, every Goodstein
sequence converges to 0.
The secret underlying Goodstein's theorem is that the hereditary representation of 
in base 
mimics an ordinal notation for ordinals less than some number.
For such ordinals, the base bumping operation leaves the ordinal fixed whereas the
subtraction of one decreases the ordinal. But these ordinals are well ordered, and
this allows us to conclude that a Goodstein sequence eventually converges to zero.
Amazingly, Paris and Kirby showed in 1982 that Goodstein's theorem is not provable in ordinary Peano arithmetic (Borwein and Bailey 2003, p. 35).
