Let 
be the set of probability measures 
in 
with 
.
According to
facts 1 and 2 above the mle 
belongs to
for all large 
almost surely on the explosion set.
Let 
.
and 
.
Thus for all large either
or 
.
We will show that
almost surely on the explosion set. This would show that
for all large on the explosion set, establishing the
theorem.