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The Brouwer fixed-point theorem says that every continuous map $T:{\overline{B}}_R(0)\to {\overline{B}}_R(0)$ has a fixed point. To use the supplied outward vector field zero theorem, first extend the construction to all of ${\mathbb{R}}^n$. LetThe radial retraction onto a closed ball $P_R$ is continuous, so $g$ is a continuous vector field on the required whole space. On $|x|=R$, the Cauchy-Schwarz inequality givesThe supplied theorem gives $g(x_0)=0$ with $|x_0|\le R$. There $P_R(x_0)=x_0$, hence $\boxed{T(x_0)=x_0}$. This explicit extension avoids presuming that the original map $T$ was defined outside the ball.

The Brouwer fixed-point theorem says that every continuous self-map of a closed ball in ${\mathbb{R}}^n$ has a fixed point.

Suppose $g$ has no zero in ${\overline{B}}_R$. Thenis a continuous map from ${\overline{B}}_R$ into its boundary, hence into ${\overline{B}}_R$. The Brouwer fixed-point theorem gives $T(x_0)=x_0$. This forces $|x_0|=R$ andcontradicting the assumed boundary inequality. Therefore $\boxed{g(x_0)=0\text{ for some }|x_0|\le R}$. This is the outward vector field zero theorem.