Theorem 3
No orbit of the Gauss map has a Lyapunov exponent smaller
than 
.
Proof
Let 
be any initial
point in 
such that 
exists. We will show
that the product 
which appears in
the definition of 
must be at least 
(for
N sufficiently large) which will prove the theorem. We consider
two subsequent elements 
and 
of the orbit
of 
. If k=N, enlarge the product by one term. Note 
and 
are related by 
.
If 
then the contribution of 
to
the product is at least 
. If instead 
then 
so
the contribution of 
to the product
is at least 
. This proves the theorem.