Before we proceed with some proves relevant for our stochastic processes we need to see and understand two important proves, which are stated as Lemmas (Lemma 1.1 (a) and Lemma 1.1 (b)) below.

Lemma 1.1 (a)

If  converges (i.e., ) then it implies (i.e., )

Proof of Lemma 1.1 (a)

What we will prove here is  (see carefully this is what we have to prove as written above)
Since  converges then for any  one can find  such that  holds true for all values of , then choose that N and write

Now for  we have

, where

 being sufficiently close to 1 we have
 [this can be arbitrarily done considering any combination of  and , such that  can be made lesser and lesser to ].

Now we need to find the second term which is , hence we have

Look carefully and we immediately note that the first term is bounded by , while the second term is bounded by . Hence we have

Combining these two we have , provided  being sufficiently close to 1.