Sustainable growth in the long run
As I said in the comments, without additional details proving anything is either impossible or trivial. Impossible if I do not make any assumptions, trivial if I am allowed to make any assumption I like.

Let me give you a hint on how to solve this in the framework how it is usually taught.

The assumptions made are:


There are decreasing returns to scale to any single production factor
There are constant returns to scale to all production factors together
Technology and the labour force grow at an exogenous rate (which may be 0)
The production function is cobb-douglas form


Then, denoting national income at time $t$ as $Y_{t}$, capital as $K_{t}$, labour as $L_{t}$, technology as $A_{t}$, a fixed factor such as land as $X$ (which is independent of time) and a non-renewable resource extraction as $E_{t}$ we can formulate the production function as:

$Y_{t}=A_{t}K_{t}^{\alpha}X^{\beta}E_{t}^{\eta}L_{t}^{1-(\alpha+\beta+\eta)}$ 

We can determine the income per capita (or more precisely the income per labourer) by dividing both sides by $L_{t}$. Using small letters to denote a variable in per labourer terms we get:

$y_{t}=A_{t}k_{t}^{\alpha}x_{t}^{\beta}e_{t}^{\eta}$

Note that whereas $X$ is independent of time, $x$ is not because the labour force grows over time so the amount of land per labourer changes over time. You can now rewrite this equation in terms of growth rates to determine how it is (im)possible to continue to grow income per capita even in the presence of a non-renewable and fixed production factor.