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Let $A$ be an open set in $\mathbb{R}^n$. I would like to prove the following result:

There exists a sequence of compact sets $\{D_i\}$ with the following properties:

(a) Each $D_i$ is a subset of $A$.

(b) $D_i \subset Int(D_{i+1}) \ \forall i$, where $Int()$ denotes interior.

(c) $\bigcup_{i=1}^\infty D_i = A$.

(This result is used without proof in Lemma 16.2 of "Analysis on Manifolds" by James Munkres.)

Constructing a sequence with the first 2 properties is easy. We start with an arbitrary compact set $D_1 \subset A$ with non-empty interior (e.g. a closed ball centered at some point of A). We then use the following fact: every compact set contained inside an open set $A$ can be enclosed in the interior of another compact set contained inside $A$. This allows us to construct a sequence $\{D_i\}$ satisfying (a) and (b). However I don't know how to ensure that their union is $A$.

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  • $\begingroup$ We can write $A$ as a countable union of open balls. Can you see what to do from there? $\endgroup$ – Prototank Oct 25 '18 at 14:39
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Take$$D_n=\left\{a\in A\,\middle|\,\|a\|\leqslant n\wedge d(a,\partial A)\geqslant\frac1n\right\}.$$

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