Looking at the answers given, I think they may be missing a key distinction made by Aristotle and Scholastic philosophers (who were the ones most fond of reducing fallacies to infinite regresses).
Scholastic philosophy (in particular St. Thomas Aquinas) makes a careful distinction between what they call per se causes and per accidens causes.
I think the distinction is valid. I will illustrate with examples. A “cause” in Scholastic parlance is anything on which something depends, either for its very existence, or else for coming to be the way it is now.
Strictly speaking, or “per se,” cause and effect are always simultaneous. Let’s suppose a batter is in the process of swinging his bat at a fast ball. Right now, his bat is in contact with the baseball; therefore, right now, the batter is the per se cause of the change in velocity of the baseball, because that change depends (right now) on the action of the bat (which depends in turn on the batter).
Now lets fast-forward to a few moments later, when the baseball is flying high towards left field. At this point, the batter is no longer the per se cause of the baseball’s flight: he is merely what Aquinas would call the per accidens cause. The batter now has no influence on the trajectory of the ball: it no longer depends on him. In fact, the batter immediately begins to run toward first base, and yet the ball is completely unperturbed.
Now, returning to the problem of infinite regresses: in reality they are only impossible in a line of simultaneous per se causes. At the moment of the crack of the bat, something must be the ultimate origin of the change in velocity in the baseball. A baseball does not do random about-faces in mid-air; something must be there to apply force to it.
From a philosophical point of view, anyway, there is actually nothing that prevents an infinite regress of per accidens causes. For example, there is nothing (philosophically speaking) that prevents an infinite succession of living creatures. Why not? Because each living being depends on its parents (or parent) only per accidens: at the moment of its generation, it depended on them per se, but it does so longer.
(St. Thomas Aquinas says as much in his short treatise De principiis naturae (On the Principles of Nature), famously disagreeing with St. Bonaventure. I think Aquinas is right on this point, thanks to his distinction between per se and per accidens causes.)
Applied to the original question regarding time, in my opinion you cannot prove that time has a beginning by attempting to reduce it to an infinite regress.