Seems to me you should define rich, because the term is a moving target and most people 100 years ago would consider nearly every American alive today as "rich." The point is, the total amount of wealth is growing and has been since we started trading with each other.
Worrying about "running out of wealth" is like worrying about the sun running out of hydrogen... pointless. There is no reason whatsoever to believe that (unless of a global economic catastrophe unlike any other in history) wealth will stop growing, much less get smaller.
Why is it pointless? Isn't wealth the amount of resources you have? Land, fresh water, animals, trees, quality air, etc. If those supplies are diminishing, what makes you think wealth is increasing?