The quest may be for references but I want to know if there is a simple way to prove the Brouwer fixed point theorem!

That is if a function $f:\bar{B}\to\bar{B}$ is continuous then $f$ admits one fixed point!

The answer for the $R¹$ is a real analysis exercise and it also holds for any interval compact!

The proof is related with Schauder fixed-point theorem?

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    $\begingroup$ It's false for $\mathbb R$. Take $f(x)=x+1$. It's certainly true for compact intervals though. $\endgroup$ – Cheerful Parsnip Mar 28 '12 at 1:11
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    $\begingroup$ yeah the interval has to be compact, sorry for the distraction! $\endgroup$ – André Lima Mar 28 '12 at 1:12
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    $\begingroup$ You like to use exclamation points! :) $\endgroup$ – Cheerful Parsnip Mar 28 '12 at 1:13
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    $\begingroup$ The number of exclamation marks probably means that you're very excited! So I'm sure you have looked at the extensive Wikipedia page! Why do the references and proofs given there not answer your question? @Jim: och, you beat me to it! :) $\endgroup$ – t.b. Mar 28 '12 at 1:13

There is a book proof using Sperner's Lemma which is (very!) elegant and (IMO) simple.

  • $\begingroup$ This one is nice! Fast answer ;-) $\endgroup$ – dtldarek Mar 28 '12 at 1:20
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    $\begingroup$ @Aryabhata: you are missing an exclamation mark somewhere! $\endgroup$ – Mariano Suárez-Álvarez Mar 28 '12 at 1:23
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    $\begingroup$ @MarianoSuárez-Alvarez: Thanks! Corrected :-) $\endgroup$ – Aryabhata Mar 28 '12 at 1:26

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