Goodstein's theorem can be proved ( using techniques outside Peano arithmetic, see below ) as follows : Given a Goodstein sequence " G " ( " m " ), we construct a parallel sequence " P " ( " m " ) of ordinal numbers which is strictly decreasing and terminates.
22.
Which means that the derivative is always negative for all n which means that our original function g _ n ( x ) is a strictly decreasing function of n which means that the first function f _ n ( x ) = e ^ { g _ n ( x ) } is also a strictly decreasing function of n because the exponential is a continuous increasing function and hence it preserves the inequality . talk ) 01 : 49, 1 June 2008 ( UTC)
23.
Which means that the derivative is always negative for all n which means that our original function g _ n ( x ) is a strictly decreasing function of n which means that the first function f _ n ( x ) = e ^ { g _ n ( x ) } is also a strictly decreasing function of n because the exponential is a continuous increasing function and hence it preserves the inequality . talk ) 01 : 49, 1 June 2008 ( UTC)
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