Proving the existence of infinite parallel universes with different versions of yourself using only undergraduate level mathematics and physics is a challenging task. The concept of parallel universes, particularly in the context of multiple versions of oneself, is speculative and lies beyond the scope of current scientific understanding. It is important to note that while there are various speculative theories and hypotheses in physics, they often require advanced mathematical and theoretical frameworks to be fully explored.
That being said, I can provide a brief overview of a speculative concept related to parallel universes known as the "many-worlds interpretation" of quantum mechanics, which is sometimes discussed in undergraduate physics courses. However, it is crucial to understand that this interpretation is just one of many possible interpretations, and it remains a topic of debate among physicists.
According to the many-worlds interpretation, every quantum event gives rise to multiple branching universes, each representing a different outcome. In this view, all possible outcomes are realized in separate parallel universes, including variations of yourself.
However, it's important to note that the many-worlds interpretation is not universally accepted, and it has its critics and alternative interpretations within the scientific community. Proving the existence of these parallel universes, especially with variations of oneself, would require empirical evidence that currently lies beyond our technological capabilities.
Therefore, at the undergraduate level of mathematics and physics, it is not possible to provide a rigorous proof of the existence of infinite parallel universes with different versions of oneself. These ideas are currently speculative and often explored within the realm of theoretical physics and advanced mathematical frameworks.