I'm not convinced we can provably show that the resources on Earth aren't capable of sustaining a great many more people, albeit with a significantly different style of living.
Of course we can. 90% of food production is based on fossil fuel inputs. Fossil fuels flow rates have plateaued and will move to decline. Because of the increased demand of population that decline will be rapid and increase quickly. When that happens massive amounts of people will starve. This will create huge social dislocations which make the maintenance of a complex system capable of space exploration unlikely.
But more basically you can add up energy consumption necessary to sustain any basic average standard of living sufficient to also maintain a functioning global economy, multiply it by population and compare it to the NOT proven reserves but net recoverable energy taking into account EROEI and it's not a good number. Google is your friend.
Further, this ignores other essentials such as topsoil depletion, fresh water and NPK inputs - NPK inputs are very limited in quantity, depleting rapidly and incredibly energy intensive to recover.
Exponential growth can't continue forever in nature.
No it doesn't assume no innovation. It's based on the fact that wishing for a magical elixir which will replace our dwindling resources is just that - wishful thinking.