I believe your reasoning follows from a set of bad premises.
No one in modern macroeconomics believe that economic growth is resource-dependent (a so-called Malthusian model, formulated around the turn of the 19th century). Since approximately the mid-1930's, economists have lived in an exogenous-growth world, formalized conceptually by the Solow model; growth comes from paramters outside the model. That is to say, holding other factors constant (the supply of labor, capital, etcetera) we still observe output growth.
To put another way, economic growth is not resource-dependent. Ergo, any finite supply of inputs is sufficient by definition to produce an infinite pattern of long-run economic growth. Why this is true is the subject of considerable debate - the general consensus is that technology increases the efficiency of factors of production, but technology alone is not sufficient to explain all observed delta in productivity. See
http://en.wikipedia.org/wiki/Productivity.
The empirical evidence is certain. Altering internal factors (savings rates, capital production rates, population rates, etcetera) can produce disequilbria in the model, but in the long-run the observed data converges on a steady-state of exogenous growth consistent with the rate of change in productivity. This is a post-industrial phenomenon, of course; Malthusian resource-dependent growth models were sufficient to explain observed economies for the great majority of human history. But no longer.