CO2 makes up 390 ppm (0.039%)* of the atmosphere, how can such a small amount be important? Saying that CO2 is "only a trace gas" is like saying that arsenic is "only" a trace water contaminant. Small amounts of very active substances can cause large effects.
Although percentage is a convenient way to talk about the amount of gas in the atmosphere, it only tells how much is there relative to everything else; percentage doesn’t give an absolute amount.
For example, you have trouble breathing on top of Mount Everest even though the atmosphere still contains 21% oxygen just like at sea level. The percentage isn't important, you need a certain number of oxygen molecules with each breath, regardless of how much or little they are diluted by inert gases. At an altitude of 8000 m the whole atmosphere is diluted.
The total number of CO2 molecules above our heads in the atmosphere is more important than their percentage in the atmosphere. If the amount of inert nitrogen gas (N2) in the atmosphere were to be cut in half then the percentage of CO2 would jump (to about 600 ppm; 0.06%) without a change in the absolute amount of CO2 and no substantial change in the energy balance of the Earth. Adding a huge number of energy-absorbing CO2 molecules to the atmosphere doesn’t change its percent number very much, only because it's being added to a vast inert N2 background.
We know the amount of CO2 in the atmosphere has increased because we have measured it. We know the climate has warmed from current and historical data. The link between increasing greenhouse gases and increasing temperature is clear: just as ink makes water more colored, CO2 makes the atmosphere more absorbing. The extra CO2 in our atmosphere is trapping energy that would otherwise escape to space. The measured global warming matches closely with the amount of energy trapped from the greenhouse gases added to the atmosphere.
A doubling of the trace molecule CO2 from 280 ppm to 560 ppm is still a trace, but just like with arsenic, the difference between a small trace and a larger trace is fatal.
* To convert ppm to percentage divide by 10,000.
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