### 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.