This is the necessary condition. Now we need to concentrate on the sufficient condition prove.
Assume the  state is transient, i.e., . Using the two stated facts given below :

(i)	if  converges then

and

(ii)

for

we can infer .Again utilizing the fact that if  and , then  which leads us to  being true. On seeing this we can immediately conclude that it contradicts our hypothesis based on which we started, i.e.,  state is transient. Hence  state is not transient.