I apologize in advance for starting yet another thread, but I would really like to have some feedback on this one.(adsbygoogle = window.adsbygoogle || []).push({});

I have a "proof" of the theorem, which surely is wrong because it relaxes one of the conditions. Start with the "official" statement:

Let X be a compact metric space, f:X->X, and d(f(x),f(y)) < d(x,y) (or alternatively, d(f(x),f(y)) < a*d(x,y), 0<a<1) when x /= y. Then there exists a (unique) point x in X such that x=f(x).

Pf (of existence): Define x_n+1 = f(x_n). Since X is compact, the sequence {x_n} contains a convergent subsequence, say {x_n_k} with x_n_k --> x and x is in X. This implies that f(x) is also in X. d(f(x),f(y)) < d(x,y) implies that f is (uniformly) continuous, so x_n_k+1 = f(x_n_k) --> f(x). The limit point thus has the desired property.

Surely, this is wrong as the only condition I need is continuity of f. But somehow, I cannot figure out which step(s) is wrong.

Any help greatly appreciated.

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# Fiexed point theorem for contractions

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