We write:

where

Now note that M represents the following digraph: 1->2->3->...->n . When we exponentiate M (remember that

is the number of paths of lenght m from i to j ) we get:

(1) (when I say dimension I am talking about the dimension of our matrices) - From here it is clear that

. (2)

Where does this come in handy? Well:

(

)

Consider

then - by (2) -

Then

but, look back at (1)

if our entry here is not 0 already, it is only one of the terms of the sum , i.e.

for some

and this clearly tends to 0 since

is fixed ;

and