The very simple, logical evidence for life as a cycle has been in front of us all along. When you are born you are coming out of a state of unconsciousness. Our “initial” state then (for this example) is that of “unconscious”, or “off”. After birth our state is set to “on”. At death we move BACK TO OFF. Since obviously we can’t have any experience of non-experience, the very next thing we’ll ever experience is another birth, or another movement into the “on” state.
I wonder if people would think differently about how they spend their time on earth if they realized that the very existence and continuation of life anywhere could lead to their own reincarnation in the future. Looking at it the opposite way – if there is no life that exists after your death, there is no life for you to experience in any capacity. The extinction of all life would be the only true death.