Proof of The $\delta$-Interior of an Open Subset of $\mathbb{R}^n$ is Open

Topological notions refer to the topology of the sets open in $(\mathbb{R}^n,d)$, a topology by Metric Open Sets Form a Topology. By claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ we have $d(z,w)=\lVert z-w\rVert$ throughout, and $B(z,r)$ denotes the open ball, which is open by Open Ball in a Metric Space is Open.

Claim 1. Let $x\in\Omega^{\delta}$ and put $K=\bar B(x,\delta)$, so that $K\subseteq\Omega$; the set $K$ is compact by claim 2 of A Closed Euclidean Ball is Convex and Compact, and $x\in K$, so $K$ is nonempty.

Let $z\in K$. Since $z\in\Omega$ and $\Omega$ is open there is a real $s_z>0$ with $B(z,s_z)\subseteq\Omega$; put $r_z=s_z/2$, so that $r_z>0$ and $B(z,2r_z)\subseteq\Omega$. The open balls $B(z,r_z)$, for $z\in K$, form a cover of $K$ by open subsets of $\mathbb{R}^n$, since $z\in B(z,r_z)$ for each $z$. By Compact Subset Criterion via Open Covers in the Ambient Space finitely many of them cover $K$: there are $z_1,\dots,z_N\in K$ with $N\ge1$ and

Let $\eta$ be the smallest of the finitely many positive real numbers $r_{z_1},\dots,r_{z_N}$; then $\eta>0$.

We show $\bar B(x,\delta+\eta)\subseteq\Omega$. Let $w\in\mathbb{R}^n$ with $t=\lVert w-x\rVert\le\delta+\eta$. If $t\le\delta$ then $w\in K\subseteq\Omega$. Otherwise $t>\delta>0$; put

By claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $\lVert z-x\rVert=(\delta/t)\,t=\delta$, so $z\in K$; and since $w-z=(1-\delta/t)(w-x)$ with $1-\delta/t>0$, the same claim gives $\lVert w-z\rVert=(1-\delta/t)\,t=t-\delta\le\eta$. Choose $i$ with $z\in B(z_i,r_{z_i})$. By claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$,

so $w\in B(z_i,2r_{z_i})\subseteq\Omega$. Hence $\bar B(x,\delta+\eta)\subseteq\Omega$, which proves claim 1.

Claim 2. Let $x\in\Omega^{\delta}$ and let $\eta>0$ be as in claim 1. Let $x'\in B(x,\eta)$, so $\lVert x'-x\rVert<\eta$, and let $y\in\bar B(x',\delta)$. By claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$,

so $y\in\bar B(x,\delta+\eta)\subseteq\Omega$. Thus $\bar B(x',\delta)\subseteq\Omega$, that is, $x'\in\Omega^{\delta}$. Hence $B(x,\eta)\subseteq\Omega^{\delta}$, and as $x\in\Omega^{\delta}$ was arbitrary, $\Omega^{\delta}$ is open in $(\mathbb{R}^n,d)$ by Open Subset of a Metric Space.