Suppose that 
for all t. If I is an interval
such that |I|>B then any infinite orbit intersects I.
The sum of 
over the points in I lying in a given infinite
orbit is bounded above by I and below by |I|-2B.
Figure 6: One orbit
If I is large enough then the sum of 
for 
can
be made arbitrarily close to the number of infinite orbits of f;
the singleton orbits don't contribute since 
for
those orbits. Thus in the limit the average of df over an
interval 
of consecutive integers must
become arbitrarily close to the number of infinite orbits of
the permutation. 
