# Fixed point theorem

#### Fermat

##### Active member
Schauder Fixed Point Theorem

In particular, where it restricts g to con(F). I don't understand how we know con(F) is a subset of C. C is convex so if C contained F, then certainly con(F) would be a subset of C. But how do we know F is a subset of C.

Thanks

#### Opalg

##### MHB Oldtimer
Staff member
That is a good point, and it looks to me as though that link skates over this bit of the proof. What it should say is that, because $K = \overline{f(C)}$ is compact, for each $\varepsilon>0$ there exists a finite $\varepsilon$-net $F = \{x_1,\ldots,x_n\}$ for $K$, whose elements $x_1,\ldots,x_n$ are all in $f(C)$. It is easy to show that such an $\varepsilon$-net exists (because the set of $\varepsilon$-neighbourhoods of all points in $f(C)$ is an open cover of $K$ and therefore has a finite subcover). Since $f$ maps $C$ into $C$, it then follows that $\mathrm{con}\,F\subset C.$