Finite Infinity Paradox

I have been giving some more thought about my previous post where I stated my belief that the universe is infinite. It has occurred to me that there is a problem  here and it is this: no matter how distant two objects might be apart in the universe, the space between them can only ever be finite. Similarly, if we consider two objects which start off touching, and then start moving away from each other, in order for them to be an infinite distance from each other they need to continue moving away for an infinite length of time. However, this cannot happen because no matter how much time goes by, it will always be finite and so will the distance separating the two objects. It would therefore follow (call this McAdam's Infinity Paradox if you like - I have not read these ideas expressed anywhere else) that in an infinite universe physical objects may only approach a separation of infinite distance, but never reach it.

We now find ourselves in an infinite universe where all distances can only ever be finite.

Perhaps the idea of infinity should be viewed as a concept rather than a physical reality, and that spacial distances are actually an illusion because of our own particular limitations in our perceptive ability. Consider that two particles on the atomic scale which are super-entangled will affect each other instantly and independent of the speed of light no matter how great the separation between them. In what sense can we say that there is a spacial distance between the two particles given that they act as if there is not? Could distance itself be an illusion?