The only proof you're going to get is modeling. That is a tough pill to swallow because the most accurate measurements have happened over the last few decades and the ice ages were like 100,000 years and inter-glacial periods are something like 10,000 to 15,000 years. The information we've gleaned from ice cores and such can have a range of error however bounded it is.
1. The models we have are skillful, that means they have predicted 10 or 15 years ago what today looks like with a good degree of accuracy.
2. We know man is contributing CO2 to the atmosphere.
3. We know CO2 is a GHG.
4. We wondered why CO2 wasn't rising as fast as we calculated it should, found it is in the ocean.
5. Historical records indicate that atmospheric CO2 lags the warming trend, but the amount of warming cannot be explained by the earths orbit around the sun. CO2 provides a feedback that pushes the trend. Vostok ice core.
6. The sun is cooling, has been for a while, yet temperatures are still rising.
7. CO2 has never been this high or risen this fast, and there has been so little time for temperature to react to CO2 as a GHG.
8. We have been working on the CO2 GHG theory for over 100 years "theory was first proposed by Swedish scientist Svante Arrhenius in 1896"
9. Nearly every scientist that worked on GHG theory agreed increasing GHG would cause the global average temperature to rise. "What they don’t agree on is by how much."
10. Measured radiance from earth in the wavelengths of CO2, CH4, and O3 have decreased and downward radiance for CO2(and CH4, O3) has increased by the same amount. -> CO2 trapping heat.
There's still plenty we do not know.
1. When will Antarctica really start melting? The Arctic is predicted to be ice free in summer months before 2030.
2. Why are ice crystals forming on the underside of a glacier in the Antarctic?
3. What effect, if any, do brine flows at the bottom of the ocean have?
4. When will the oceanic "conveyor belt" stop, what will that do?
5. How much will climate warm by?
6. How fast will sea levels rise, and by how much?
7. How will our models improve as we move forward?
With the information we do have, it is logical to believe that CO2 will be the forcing mechanism that starts the warming trend.