I have to honestly say that no one thing alone has ever convinced me of anything. It is, rather, many things combined that have provided a clear parallax view of things that would normally seem impossible. My personal trust that Reincarnation is a real process comes from sources too numerous to detail, but I can say that it is a combination of logic, knowledge and experience with life, anecdotes and even science.
Logic implies that, one life alone is too short an experience to give any meaning to life. Knowledge and experience with life has taught me that people are sometimes able to know things they were never taught during this lifetime. Anecdotal case histories by Dr Ian Stevenson, Carol Bowman and others are too credible to be ignored. And, scientific research has provided too much evidence of extrasensory phenomenon which provide a healthy foundation for Reincarnation.