Stellar evolution is the life cycle of a star. A cycle suggests that upon death of a star is born. How is this possible?

When stars (our sun) die, they become red giants and may even go supernova. Supernovas form into large clouds of gas called nebulas. Nebulas are the stellar nurseries for stars. The slightest peturbation in a nebula (likely the shockwave from a nearby supernova) may cause the gas in a nebula to begin collapsing and condensing into a tight ball. When that ball becomes dense enough to fuse hydrogen, a new star is born, and hence the cycle continues. Prior to our sun there was a nebula, and prior to that nebula, there was another star; the ancestor of our sun.