The radius of a graph section is bounded on a compact boundary collar

With the standard step function of F4, put $\phi (t)=R_e+(t-R_e)\sigma (t-R_e)$ and $f=\phi \circ r$ on ${\mathrm{\Sigma }}^{\circ }$. Then $\phi$ is smooth, constant for $t\le R_e$, and equal to $t$ for $t\ge R_e+1$. Its derivative ${\phi }^{\mathrm{\prime }}(t)=\sigma (t-R_e)+(t-R_e){\sigma }^{\mathrm{\prime }}(t-R_e)$ is bounded, since ${\sigma }^{\mathrm{\prime }}$ vanishes outside $[0,1]$ and is continuous on that compact interval. By the chain rule, $df={\phi }^{\mathrm{\prime }}(r)dr$. Its norm is bounded on the far end by the hypothesis, zero near $r=1$, and bounded on the transition annulus because $G(\{R_e\le \mathrm{\mid }x\mathrm{\mid }\le R_e+1\})$ is compact in ${\mathrm{\Sigma }}^{\circ }$ and $h$ is smooth and positive there. Thus $\mathrm{\mid }df{\mathrm{\mid }}_h\le L$ throughout ${\mathrm{\Sigma }}^{\circ }$ for some finite $L\ge 0$.

Let $\gamma$ be a piecewise smooth interior path from $y$ to $z$. On each smooth piece with parameter interval $[a,b]$, the chain rule and the covector norm give $\mathrm{\mid }(f\circ \gamma {)}^{\mathrm{\prime }}(t)\mathrm{\mid }=\mathrm{\mid }d{f}_{\gamma (t)}(\dot{\gamma }(t))\mathrm{\mid }\le L\mathrm{\mid }\dot{\gamma }(t){\mathrm{\mid }}_h$. Both derivative and speed are continuous on the closed piece, so F6 applies. Newton–Leibniz, monotonicity, and linearity give $-L{\int }_a^b\mathrm{\mid }\dot{\gamma }{\mathrm{\mid }}_{h}dt\le f(\gamma (b))-f(\gamma (a))\le L{\int }_a^b\mathrm{\mid }\dot{\gamma }{\mathrm{\mid }}_{h}dt$. Taking absolute values, summing the finitely many endpoint increments, and using the length definition yields $\mathrm{\mid }f(y)-f(z)\mathrm{\mid }\le L{\mathrm{length}}_h(\gamma )$.

On the fixed collar, smoothness and compactness bound the lengths of collar-coordinate segments from $(s,u)$ to $(s,a)$. On the compact connected outer collar face, paths from a fixed point to its other points have uniformly bounded lengths, using finitely many coordinate charts (for multiple boundary components use one fixed point on each, finitely many since $S$ is compact). Concatenating gives a uniform finite path-length bound to a finite collection of fixed interior points for every collar point with $u>0$. Step 2.1 then bounds $f$ there, and hence bounds $r$ since $f=r$ at large $r$. This uses only smooth collar geometry, not boundary extension of stationary graph coordinates.