Our 'economy' represents our activity in life, our need for housing, food, transport, mining, logging, fishing, farming, making money, etc, etc.
Yes, and economic growth is growth in GDP, measured in dollars.
And as the planet and its resources, arable land, etc, is finite, perpetual growth is not possible.
That follows ONLY if you cannot increase GDP without increasing resource use. Can you demonstrate that this is true?
(Hint: You cannot, because it's not true).
Buying and downloading new software for your computer is one example of an economic activity which increases GDP, but which requires no additional resource use to achieve. High speed share trading is another example.
Thinking is an economic activity (if you can find someone to pay you to do it). Most jobs, and all really high-paid jobs, in the developed world are mostly thinking.
A person needs the exact same resources to live, whether he thinks or not.
Given that there are ways to increase GDP with no increase in resource use, your claim "perpetual growth is not possible" is demonstrably untrue of
economic growth.
It doesn't matter that it is true of
population growth or of
resource use growth. Those things can only be relevant if they are always and unavoidably required for GDP growth, and at least since the Industrial Revolution, they have not been.
Economic growth has massively outstripped both population and resource use growth for at least two centuries now, and the gap between them is rapidly widening as we move to an information economy.
Your belief that economic activity requires the use of natural resources is simply false.