CO2 and the other greenhouse gases make up only a small part of our atmosphere.  How can they possibly have that great an effect?

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Be grateful for small favors.

GHGs are indeed trace, being measured in parts per million and parts per billion. That, however is enough. 180 to 300 ppm has been adequate to keep our earth at a comfortable living temperature, rather than the icy world that would be expected otherwise.

The higher levels correlate with higher temperatures both historically, and currently.

Ours sister planet, Venus, which has an atmosphere that is primarily carbon dioxide, holds huge amounts of heat in the atmosphere and, not that anyone has tried it, we are told lead would melt on its surface.