Sequence of continuous functions that approximate $\chi_K$ The following is from Schilling's Brownian Motion.
Let $K \subset \mathbb{R}^d$ be compact. Then $U_n = K + B(0,1/n)$ is open and 
$$u_n(x):= \frac{d(x,U_n^c)}{d(x,K)+d(x,U_n^c)}$$ is a sequence of continuous  functions. In this case, why do we get for $r \in (0,1)$
$$u_n(y)=\frac{d(y,U_n^c)}{d(y,K)+d(y,U_n^c)} \ge \frac{(1-r)/n}{1/n} = 1-r. \; \forall y \in K+B(0,r/n)$$? I don't see how this makes sense. If $y \in K$, then $d(y,K)=0$ so we should get $u_n(y)=1$. But how do  I consider this for other cases?
Hence, we have $d(y,K)+d(y,U_n^c) \le 1/n$.
Below is the full proof given by Schilling, where $\chi_n$ is a cut-off function that integrates to $1$.

 A: (I). For any $s>0$: For $x\in K$ we have $B(x,s)=\{x\}+B(0,s)\subset K+B(0,s) .$ And if $y\in K+B(0,s)$ then $y=x+z$ for some $x\in K$ and some $z\in B(0,s)$ so $y\in B(x,s).$ Therefore $$K+B(0,s)=\cup_{x\in K}B(x,s).$$ 
(II). For $r\in (0,1)$ and $y\in K+B(0,r/n)$ we have  $y\in \cup_{x\in K}B(x,r/n)$ so $y\in B(x',r/n)$ for some $x'\in K.$ 
And $B(x',1/n)\subset U_n,$ so for any $z\in U_n^c$ we have $d(x',z)\geq 1/n.$ By the triangle inequality we have $$d(y,z)\geq  d(x',z)-d(x',y)\geq 1/n-d(y,x)>1/n-r/n=(1-r)/n$$ for any $z\in U_n^c.$ Therefore $$d(y,U_n^c)\geq (1-r)/n.$$ On the other hand $$d(y,K)\leq d(y,x')<r/n.$$ For brevity let $A=d(y,U_n^c)$ and $B=d(y,K).$ So $A\geq (1-r)/n$ and $B\leq r/n.$ We have $$u_n(y)\geq 1-r\iff \frac {A}{A+B}\geq 1-r\iff$$ $$\iff  A\geq (A+B)(1-r)\iff rA\geq (1-r)B.$$ Now observe that $rA\geq r(1-r)/n$ and that $(1-r)(r/n)\geq (1-r)B.$
A: I believe the bound $\mathrm{d}(y,K) + \mathrm{d}(y,U_n^{\mathrm{c}}) \leq 1/n$ is wrong without any other assumption. Indeed, take $K=\bar{B}(0,1)\setminus B(0,1/(2n))$ and $y$ in the hole, i.e. $y\in B(0,1/(2n))\setminus B(0,(1-2r)/(2n))$, where $r\in (0,1/2)$. Then we have $U_n=B(0,1+1/n)$ so $\mathrm{d}(y,U_n^c)\geq 1+1/(2n)>1/n$.
A: Indeed, if $y\in K$ then $u_n(y) = 1 \geq 1-r$.
The estimate is non trivial in the case $0 < d(y,K) < r/n$.
