In the beginning there was nothing, or whatever. Then there was time. No wait. What? How can time begin, if there is no time at which it could have begun (because it is time itself)?
We can only reason that time must have no beginning. It was always there.
Yes, specific things may be happening at specific times. However, so long as time continues, more possibilities will occur. And if time never stops, eventually it will formally undergo each possibility at some location at some point.
But we didn't prove time has no end. We proved it doesn't have a beginning. Fortunate for us, we can then reason that every possibility must have already undergone the formality of actually occurring :-)
Okay, so you've proved that time's infinite, as it has no beginning. But there are also (probably) infinite possibilities, so we have to decide which is 'more infinite' or 'which infinity is greater', à la Castor. There is no actual answer to this question because the two infinities are incomparable- one variable is space AND time (possibilities rely on time and space) and one variable is solely time. If you could prove whether time was more or less infinite than space (which would be tough, as they're both (supposedly) linear) you could answer this question.