I have refined some of the expression of the argument above now. That is an argument from logic and should convince all rational people. It rests completely on the foundation of the first premise. If you deny the self, such as Buddhists do, then you deny the conclusion too. If you accept there is a self, then you are bound to the conclusion of reincarnation logically.
Now I will argue from theism. If one accepts the worldview that god the creator created this universe for us souls and Earth is like a test for us to do good deeds or prove our loyalty to god so that we may get eternal life with him or paradise, then one must still accept reincarnation or be forced to accept that god is unjust, and cruel, uncaring or irresponsible.
This is because nobody is born equal on this planet. Everybody is born with different bodies, in different circumstances, to different parents. Some are born as physically strong and healthy, others are born weak and unhealthy, or with disabilities(blindness, deafness etc) Some are born ntelligent, some are born dull/stupid/retarded. Some are born with attractive features, some are born with ugly features.
Everybody has a different life, different circumstances, different opportunities and different life spans. Some lead a comfortable life, enjoying good health, wealth and success, while others lead a life of suffering, bad health and failure. Some have to live through horrendous wars, while others never see a war in their life. Some are treated like gods, while others are treated like slaves. Some die as soon as they are born, while others live to 120.
It is easy for anybody to see how unfair this set-up is. If this world was a test for us - his children - then it would only be fair that god made everybody equal and gave them equal opportunities, leaving us to make our own choices and allowing there to be a fair system of reward and punishment. However, if he creates one person who is naturally very aggressive and another who is naturally passive, then how would it be fair to condemn the person who was created naturally aggressive because they committed aggressive acts? If he gives one person a lifespan of 20, and another a lifespan of 100, then obviously one gets less opportunity and chances to redeem themselves than the other. If one is born in poverty and another is born in riches, then how would it be fair to condemn the person who was born poor because they stole?
Simply put, such a god is most definitely not deserving of worship, even if he did create us. In order to maintain the statement that god is just(as many theists believe) and to account for the differences in birth, death and circumstances of people is to believe reincarnation. It is because of the actions done by us in our previous life which leads to the different births, deaths and circumstances. Somebody is born a genius and another born stupid, because the one born a genius had karma to get such a birth in this life. Somebody is born healthy and another born disabled, because the disabled had done had karma to get a disability etc
So reincarnation can be proven purely through logic. It also proves to be the most consistent and satisfactory belief for one who believes in god. Not believing in reincarnation leads one to necessarily conclude god is unjust.