Why can't the Universe be eternal?

Most of us agree that the Universe may go on existing in some form for an indeterminate time into the future. In other words, it can and probably will continue forever, even if it becomes exceedingly cold and diffuse as entropy approaches infinity. Why then do we insist it can’t possibly go on infinitely long into the past? If you have an argument as to why an infinitely old Universe is impossible, make sure it’s a solid one, not something based on hand waving, sophistry, or appeals to “common sense” since common sense says nothing remotely like this Universe is possible in the beginning. Avoid logical fallacies and prove to me the Universe had a finite beginning but can continue infinitely far into the future.