well....1) If time has a beginning does this prove there is such a thing as eternity?
2) If the universe is expanding and will eventually be destroyed will time cease to exist when this happens and perhaps be proof of eternity?
1."Time"..is a man made concept...it is not universally accepted (unless E.T.'s land wearing watches).
2. Scientist say that the universe is expanding exponentially yet slowing, which leads some scientist to think that, like a rubber band or something else elastic, it will eventually start to recede and over a period of "time" would eventually return to its original point of origin leading to another big bang, if this is the case then you might think that its a never ending process (eternal) of expand and contract, big bang, planets form, life begins, universe contracts start all over again.
I would say no, it would not be eternal. when anything is created or destroyed things are lost in this process (you can only burn something so many time before there is nothing left to burn) which means that if this were the case the entire universe has a finite number of time to destroy and create itself before it no longer has the resources to do so.