Life after Death is a part of life cycle ,similar as the day after night is a part of day & night cycle.The Universe builds up and then gets destroyed ; plants grow,get desroyed again convert into seeds and back to trees.Thus we can see that each of the physical act is cyclic in nature.If we won't believe in life after death ,then the cycle remains incomplete.Now this is impossible.The cycle has to complete.The other proof can be taken as-
1. see any infant carefully. He many a time reacts to certain thought process ,which is going on his mind , as can be seen from his expressions,which are equivalent to any adult reactions These cant be reaction to any present life.Hence it can only be the result ofmemories of previous life , the memoriries of which still remain in his mind. These memories gradually subside quite quickly,as he grows.
Many a times even some grown up persons with previous life memories are found .These are found all over the world. At present also there may be a few with such memories. These memory levels very from case to case.
The above prcesses if fond true can only be explained by accepting the existance of previous life.
The explanation to the process of life after death, can be easily understood if the basic principals of Vedic phelosphy are properly understood.
The Vedic phelosophy takes three basic Things as endless (Anaadi),i.e., the one which have no begining and no end.
These are
1 Nature
2. Soul
& 3 . God
Out of these three basic things Nature is a jada padarth which cant initiate a new process on its own ,as can be seen with any worldly item like even a computor. Now God is a omnipresent force which is Chaiten and is responsible for forming this Universe from this nature,which acts as a raw material.
The soul is the end user. Soul is also chaiten and hence can initiate a new process on its own.
The very beatiful part of this Universe is that all its hidden systems can be easily understood if you only see its present working carefully. As in our daily life ,we have to have a Creator , a Raw material and a User in any of our worldly processes , similarily is the case of the Creation of this Universe, which has a God as the Creator ; Nature as the raw material and the Souls as the end users.
What I mean to say is that once you understand the Basics , the answers to all other phenomina are easily understood.
Now let's come to the original question -Life after Death. The Souls are the users of this big creation. Souls have their own existance.For this fact, to be verified ,look to any animal ,insect or any creature, one thing you can notice and you can't deny is their self esteem. None of the creature bows to other. It maitains the same even up to its end. What conclusion you would like to draw out of this, except accepting their indivuality, which never ends.Thus souls which are not a God formed material have a existance which can not have a end. and similarily it has also no begining.
Now if you have follwed and understood this, it will not be very difficult for you to realise that Souls continue to exist even though the body perishes.The rest will not be difficult to realise. The Souls continue using the creation by taking a rebirth or birth after death and death after the birth.

I think this is perfect answer .Its according to the vedas