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I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general public. The best I could think of was Goodstein sequences: if you take a particular instance of Goodstein's theorem, then the shortest proof in Peano arithmetic will be absurdly long unless the instance is very very small, but using ordinals one has a lovely short proof.

My question is this: does anyone have a more down-to-earth example? It doesn't have to be one where you can rigorously prove that using infinity hugely shortens the shortest proof. Just something where using infinity is very convenient even though the problem itself is finite. (This is related to the question asked earlier about whether finite mathematics needs the axiom of infinity, but it is not quite the same.)

A quick meta-question to add: when I finally got round to registering for this site, I lost the hard-earned reputation I had gained as a non-registered user. I am now disgraced, so to speak. Is that just my tough luck?

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    $\begingroup$ I take it by "infinity" you don't mean "one-point compactification," which is also very useful in its own right. $\endgroup$ – Qiaochu Yuan Nov 16 '09 at 23:41
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    $\begingroup$ Alakazam ! $\endgroup$ – Anton Geraschenko Nov 17 '09 at 0:54
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    $\begingroup$ See also mathoverflow.net/questions/551/… $\endgroup$ – Anton Geraschenko Nov 17 '09 at 0:56
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    $\begingroup$ If I may make a request, it would be wonderful if our colleague Gowers changed his or her display name to First Last or First M. Last. $\endgroup$ – Greg Kuperberg Nov 20 '09 at 21:35
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    $\begingroup$ Not the latter as I go by my middle name. Happy to make the change but have not managed to find where I can do it. (Please excuse my utter incompetence.) $\endgroup$ – gowers Nov 21 '09 at 16:29

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What about the puzzle named escape on Joel David Hamkins' homepage?

http://jdh.hamkins.org/escape/

That not all the squares occupied by the three stones in the initial configuration can be vacuated is shown with infinite series.

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