The proof is actually quite simple; Mathematics and Chemistry, or more generally, Physics. These are all very well known properties as solid as 1+1=2. The properties and chemistry of carbon dioxide has been well known for a very long time (since 1824), but the effect of adding carbon dioxide to our atmosphere was not seriously considered until 1962 when the Mariner 2 space probe returned data from the planet Venus. Carl Sagan, and colleagues, studied the data and discovered that Venus was far too hot for its distance from the Sun. The cause of the extra heat was also contained in the same data.  The data also revealed a high atmospheric concentration (97%) of carbon dioxide. This information was more of a wake up call than a discovery of chemical theory. The "green-house" property of atmospheric carbon dioxide was already known, just not seriously considered.  (The infrared opaqueness of atmospheric carbon dioxide was discovered by Joseph Fourier in 1824).

Since the volume of carbon dioxide emitted from our cars, factories, etc. is measurable and since the volume of our atmosphere is well documented, and since the chemical properties of carbon dioxide is well known, it is not difficult to calculate the effects. The physics is as reliable as gravity, a ruler, counting currency or dropping a Mentos in a bottle of diet coke. It is no mystery that adding $2.00 to $10.00 results in $12.00 or that adding bubble bath to your Jacuzzi will make bubbles or that adding 28.2 trillion metric tons of carbon dioxide per year to 5.2 million billion metric tons of atmosphere will make a warmer planet.

There are, indeed, natural global climate cycles, but adding the volume of carbon dioxide that we are pumping into our atmosphere makes a calculable thermal addition to the natural cycles.