It is easy to prove that the
universe is
not infinite in size. If the universe, as is almost
universally accepted, started at a finite size (actually a
point, followed by the Big Bang and/or Cosmic Inflation), and has always expanded at a
finite rate (there is no evidence to show otherwise), then it would take an infinite amount of time to reach infinity in size. Since a finite amount of time has passed since the beginning of the universe, then the universe must be finite in size.
The only way the universe could be infinite is if it started off infinitely large, or it ever had an infinite growth rate.