Thanks for your question. I’d send you here, but I take issue with your approach. The burden of proof is always on the person making a claim. If I may, I advise applying the Socratic Method and have a nice discussion (Note: Always apply the Socratic Method with their consent, don’t trick them!).
It is not your responsibility to “convince” them of their err, instead it is *their* responsibility to convince you of their claims. If I say the sky is red, it is my responsibility to show that the sky is indeed red - it is not your responsibility to disprove it. If you both agree to discuss the matter, proceed without getting emotionally panty-bunched. Hell, you could hold a long dialog that could take days, or even months.
If you are an environmentalist, you have to learn this humble, very effective, and quite easy to apply communication skill. It will serve you well through life.
Once they agree to discuss the issue, define the terms and stay hyper-focused on those things. Every once in a while paraphrase and recap the discussion - this helps clarify definitions, and it ensures that the other person feels comfortable that you are listening. So, if I’m talking about the sky being red, I’m not able to start talking about my opinion of Obama. Stay on point.
In this case, you’re discussing cycles. So, have them define it. What is a cycle? Is there evidence for cycles? Why do they believe in the science of cycles, but reject that cycles can be changed by human forcings? In other words, can natural cycles be disturbed by heavy influxes of CO2? Why or why not? Where is their evidence? Remember, they are making the claim; you are trying to learn from them. Are they choosing some scientific evidence while rejecting other scientific evidence? How is this possible? By which methodologies are they able to accept the science of cycles, yet reject the science that shows how cycles are influenced? After all, they have to point to science as their evidence for cycles. Interesting, right?
Know that you will experience breakdowns and failures while having these dialogs. That’s OK! Take a breather. Shake hands. Come back to the discussion later. It’s part of the process of learning. Try not to allow emotions to enter the discussion. Don’t get heated. Passion is a good thing, but getting angry and walking away all frustrated is a problem. Face these dips head on!
So, apply the Socratic Method when someone makes a claim. Just have fun with it. No need to be rivals.