There are three main ways in which this would be likely to happen. Two of them are only likely to happen in the presence of impending crisis while the third would be helped by crisis but can happen independently.
One way in which we might end up with a world of sustainable resources is if we have no other choice. Right now, we have a choice. We still have things like oil reserves and plenty of iron ore so we can choose to do things as we always have or we can move toward more sustainable growth. We tend, not surprisingly, to continue as we always have for the most part.
Another thing that would help bring about a more sustainable economy would be higher prices for the kinds of energy and other resources that we now use. As the population of the world grows, and as it becomes richer, more and more goods and services will be demanded. We see this already as millions of Chinese, for example, buy cars instead of using bicycles. This could drive up the prices of conventional fuels and resources and make alternatives more affordable.
Finally, there is one possibility that would not be forced on us the way these other two have been. It is possible that sustainable technologies will develop and become cheaper on their own. This is more likely to happen if businesses see a potential profit. This means that it is more likely to happen if prices for conventional things are rising. But it is already happening gradually and breakthroughs could occur sooner rather than later. If such breakthroughs lowered the costs of sustainable energy and other such things, we could move more quickly to sustainability.