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What is the best way to explain that $\Bbb{R}$ is not countable assuming that the audience is formed of people who are not mathematicians?

I ask this because these days I'm in a debate with someone about this. I have presented him the Cantor diagonal argument, but it seems he doesn't understand the proof. I'm about to give up, since I think that he does not have the mathematical knowledge necessary in order to understand the problem, but before I do that I wanted to know if you had any such 'debate' how would you proceed?

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Another video. At least the guy in the movie appears to have understood it better at the end. – arjafi Mar 12 '13 at 20:06
@Asaf: It also unfortunately reinforces the stereotype that math students are more interested in math than sex. – arjafi Mar 12 '13 at 20:18
Don't. Just... just don't. – Qiaochu Yuan Mar 12 '13 at 20:25
@Asaf: Yet this inconstancy is such / As you too shall adore; / I could not love thee, Dear, so much, / loved I not Forcing more. – Brian M. Scott Mar 12 '13 at 20:33
@Asaf: I thought that your answer was perfectly reasonable. I’ve done it a little differently, but not much. (And of course with mixed success.) – Brian M. Scott Mar 12 '13 at 20:38

You can play the devil's game with him. ( I have added this link since I could not find a better link explaining this. If someone, finds a better link feel free to edit.)

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Haha I like it. I've never seen that before. Simplistic, but fun. – Keaton Mar 12 '13 at 20:06
@AsafKaragila As a first step, a non-mathematician is not really interested in a rigorous proof for the uncountability of real numbers. I believe, he only needs an appreciation of what uncountability means. Once he gains this appreciation and if he is further interested, he will automatically make efforts to understand and appreciate the proof. – user17762 Mar 12 '13 at 20:12
In the link it is explained why $\mathbb{N},\mathbb{Z}$ and $\mathbb{Q}$ are countable. It is only mentioned that $\mathbb{R}$ is uncountable, with a reference to the Wikipedia article on Cantor's diagonal argument. – Martin Brandenburg Mar 12 '13 at 20:26

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