Fixed Point Theorem

If $g$ is a continuous function $g(x) \in [a,b]$ For All $x\in [a,b]$, then $g$ has a Fixed Point in $[a,b]$. This can be proven by noting that

$$g(a)\geq a \qquad g(b)\leq b$$

$$g(a)-a\geq 0 \qquad g(b)-b\leq 0.$$

Since $g$ is continuous, the Intermediate Value Theorem guarantees that there exists a $c\in [a,b]$ such that

$$g(c)-c=0,$$

so there must exist a $c$ such that

$$g(c)=c,$$

so there must exist a Fixed Point $\in [a,b]$.

See also Banach Fixed Point Theorem, Brouwer Fixed Point Theorem, Kakutani's Fixed Point Theorem, Lefshetz Fixed Point Formula, Lefshetz Trace Formula, Poincaré-Birkhoff Fixed Point Theorem, Schauder Fixed Point Theorem