The first lemma (page 
) state that:
Let 
be any spacetime with a simply connected Cauchy surface
. Let 
be a smooth closed compact orientable
2-surface in 
. Then no inner directed null geodesic from 
with respect to 
is part of 
.
Proof. Let t be a time function for which 
is a surface of constant
t, and let 
be the orbit of 
under diffeomorphisms
generated by 
. As 
is simply connected, the
timelike surface 
separates the spacetime into disjoint
parts, interior and exterior to the surface. Each infinite affine parameter inner-directed null
geodesic 
from 
must first
intersect 
for t > 0 at a point p. But p is in the
timelike future of 
: 
. Thus p cannot
lie on 
, and 
cannot be a generator of
.
And now the proof of the lemma page . first let us recall it.
Let M be a spacetime with a simply connected Cauchy surface 
. Let the metric be de Sitter outside of a compact set S. Let S be such that 
is not empty. Then no trapped surface 
in 
intersect 
.
Proof. Because we take the outside metric to be de Sitter we have 
which is inside 
.
This means that 
.
We do this proof by contradiction. We assume that 
is not empty.
Let 
. There exists a curve 
from 
to p.
Like 
there exists through p a null geodesics 
with an infinite length.
must be a null geodesics in 
, because if it is not then one can find a shorter curve and therefore p is not in 
. And 
has an infinite affine parameter.
This means that we have a null geodesic with an infinite lenght from a trapped surface.
This is the contradiction because, as 
is trapped surface, there must be a conjugate point within a finite affine parameter. Therefore 
cannot run to infinity.
Then 
is empty.