I don't believe scripture proves this as much as it is more consistent with the viewpoint that, when they died on the day they sinned, as scripture tells us they surely would, it was spiritual death.
Had there been no sin, I believe Adam and Eve would have grown old and died peacefully. Or possibly by an accident, especially as the population grew.1 They would not have been murdered2 or contracted HIV, but they would have died—and been none the worse for it for they would have gone to a place even better than Eden.
Francis Schaeffer, who thought an OEC view was not incompatible with the bible, speculated that pre-fall death, if it existed, was like an old dog peacefully falling asleep, and then into death, in front of cozy fire.
There are, of course, practical problems with people living forever. One would be overpopulation. About 10% of the people who ever lived are alive today. That means if nobody died the population would be at least ten times bigger—possibly much more because a huge percentage of the dead didn't make it to child bearing age.
The other is that if Adam and Eve were to live forever, they would have to live a sinless life forever. And in a twist of what I described in footnote 2, what would happen to all living, sinless descendents of Adam if he had waited, say, a few millennia before sinning? Would they be retrofitted with a sinful nature?
Anyway, this speculation proves nothing. I simply think that (a) there is overwhelming evidence of death before the fall and (b) scripture is more consistent with their promised death being spiritual rather than physical.
In this view the promise of eternal life for Adam and Eve was the same as for us: a promise of eternal life in which the present life is just a vaporous stage one. The difference was that Adam and Eve, at first, were not spiritually dead. On their own they could make choices pleasing to God. However once they, and by inheritance we, became spiritually dead we could not, prior to a second birth, please God. At that point a savior was needed.
All this is fine and dandy. I can get all my ducks in a row. There is just one problem. That mysterious old tree of life standing in the center of the garden. For when Adam and Eve are banished from the garden, we read:
Then the LORD God said, "Behold, the man has become like one of Us, knowing good and evil; and now, he might stretch out his hand, and take also from the tree of life, and eat, and live forever" (Gen. 3:22)How are we to interpret this? Superficially it sure looks like a good argument that Adam and Eve would have indeed lived forever, and so they must be denied access to the tree of life.
But I cannot see this argument as satisfactory even to those who believe Adam and Eve would have lived forever. Surely the tree isn't magic—eternal life comes from God, not from the fruit of a tree. It cannot be that the fruit was "magic" although many commentators and bible notes treat this verse as such. My bible has a footnote that argues that man was (paraphrasing) graciously preserved from the pain of living eternally in a fallen world. That doesn't smell right to me, I think the explanation is elsewhere.
Although this verse always troubled me, I have learned a simple explanation that is not inconsistent with the belief that Adam and Even would have eventually died. And even for those who believe they'd have lived forever, I think it is better than the "magic fruit" view.
We begin by pointing out that when we partake of the Lord's Supper, we use the same language as used for the tree of life. We talk of eternal life being found in the meal that we share. I believe the explanation for the tree of life in the garden is that it was a sacramental tree. It was a seal that signified eternal life—just like the bread we partake of. The tree of life was the seal—and for Adam and Eve to eat of it, after they became covenant breakers, would have been profane. Likewise, eating the bread in an unworthy manner is profane. God did not prevent them from eating of the tree of life after the fall because they would have literally lived forever. He prevented it because for them, in their fallen state, to partake of the sacrament signifying eternal life would have been obscene.
1 An interesting prob/stat problem I used to assign is to estimate our life span if all disease, including old age, was cured. It turns out to be about 500 years. In other words, given our present lifestyles, in ~500 years you have a 50/50 chance to die from some sort of random event. That makes it just like a radioactive half-life—whether you are just born or a million years old, you'd expect to live about 500 more years.
2 Or could they have been? One intriguing possibility that I have never seen discussed, but is very real if Adam and Eve would live forever if they didn’t sin, is what if one of their descendents sinned? The sin nature is inherited, we are told. Adam and Eve would then be living amongst a race of fallen creatures. How would these immortals be treated by a world of mortal sinners? Sounds like a good plot.