Lemma: Taylor bound near Z

Let $K\subset X$ be compact, and let $\varrho >0$ be such that the closed $\varrho$-neighborhood $K^{\prime }$ of $K$ is contained in $X$. Let $g\in C^{\infty }(X)$ have vanishing $k$-jet at every point of $Z$. Then for $x\in K$ with $\mathrm{dist}(x,Z)<\varrho$ and $|\alpha |\le k$,

Proof. Put $\delta =\mathrm{dist}(x,Z)<\varrho$ and choose $\delta <{\varrho }^{\prime }<\varrho$. The closed ball $\bar{B}(x,{\varrho }^{\prime })$ lies in $K^{\prime }\subset X$, so $Z\cap \bar{B}(x,{\varrho }^{\prime })$ is closed in $\bar{B}(x,{\varrho }^{\prime })$, hence compact and nonempty; a nearest point $z\in Z$ with $|x-z|=\delta$ exists, and the segment $[z,x]\subset \bar{B}(x,\delta )\subset K^{\prime }$. Taylor's formula for ${\partial }^{\alpha }g$ at $z$ to order $k-|\alpha |$, all of whose Taylor coefficients vanish, gives the integral-remainder bound stated, with the derivatives of order $k+1$ estimated on $K^{\prime }$.