This idea is not that crazy. Obviously, the Earth is finite, so all resources are finite too. But nobody ever complained that we are running out of copper, let's say, or sodium, or water, or any number of other useful things. Not because we have an infinite amount of copper inside the finite volume of the Earth, but because it is unlikely for the copper to become the bottleneck, compared to other resources.
How about oil, or natural gas? You can go to wikipedia and find the estimate of total recoverable reserves, divide by the annual consumption, and voila, we have oil only for the next 20 years, or 15, or 30, or some other number that's not very high, and certainly not expressed in hundreds or thousands of years.
Yet, this type of prediction, that we'll run out of oil in 20 years has been done for more than one hundred years. Why didn't we run out of oil?
Because proving reserves is a technical term. Oil companies are known in the industry as "oil and gas exploration and production companies". There is a whole area of banking dedicated to financing these companies. The main collateral for the loans banks advance to these companies is the reserves. The more "proven" the reserves, the more one can borrow against them.
But companies have a limited need for loans. As long as they can borrow what they need at whatever interest is economical for them, they don't have an incentive to go out of their way and prospect for all the reserves that might be on the planet. Their exploration budget is limited. So, each year they extract some oil and natural gas, and they prospect a bit more and prove more reserves. They need to have a certain ratio of probable reserves vs productive wells.
As the economy grows, the oil companies allocate accordingly more money to elevate more reserves and replenish their portfolio, so in times the proven oil reserves increase, rather than decrease.
How much longer can they keep doing this? It's not very clear, but we will not run out of oil anytime soon. Not in one century. If we stop using oil, that will be out of the concerns of climate change, not because there will be no oil left in the ground.
