How was it proven that nature is a "closed system"?
There is only one true closed system in nature and that is the earth as a whole, or Gaia.
There are three types of systems; open, closed and isolated. Open system freely exchange material and energy. Closed systems exchange only energy. Isolated systems do not exchange any material or energy.
There are many open systems through ecology. Large and small ecosystems are open to each because there are no definite physical barriers to hamper the introduction or removal of material. However, when looking at the macro view the earth is a closed system because it does not receive ecological input from space. Technically it can through meteorites, but so far it has not proven a reliable form of introduction. Meteorites are largely considered energy transfer in the closed system because they can influence the ecological system such as the mass extinction events in the past.
The closed system is proven through a number of different experiments and observations of nature. A decrease in water levels in one part of the system will equate to an increase in another part, this is known as the water cycle.
If you are looking for one person or group who proves that nature is a closed system, I'm not sure such exists. The closed system concept is a collaborative effort across many disciplines of science. If anyone deserves the credit for the concept, then it would be Randolph Clausius and William Thomson, who introduced the first and second laws of thermodynamics. The first law states matter cannot be created, nor destroyed.
This law of thermodynamics supports the idea that Earth is a closed system because the material in it stays on earth, with the obvious exception of man-made interference through space exploration.