I am reading through Munkres's Analysis on Manifolds, and I get stuck in a proof of the lemma 18.1, that is stated as following:
Lema 18.1 Let $A$ be open in $\mathbb{R}^n$; let $g:A\to \mathbb{R}^n$ be a function of class $C^1$. If the subset $E$ of $A$ has measure zero in $\mathbb{R}^n$, then $g(E)$ has measure zero in $\mathbb{R}^n$.
He made out its proof in three steps. The first and second step are mentioned in the third, where he actually prove the theorem. Let me add some pictures of the third step.
(If you need pictures of the other two steps in order to solve the question above, let me know, please)
Note: A $\delta$-neighborhood of a set $X$ is the union of all open cubes (in this case) with width $\delta>0$ and centered at $x\in X.$ The theorem 4.6 in that book states that every compact set $K$ that is contained in an open set $U\subset \mathbb{R}^n$ has a $\delta$-neighborhood contained in $U$.
So, the problem is here: When he covers the set $E_k$ by countably many cubes $D_i$ with certain properties, he asserts: Because $D_i$ has width less than $\delta$, it is contained in $C_{k+1}$.
Why this is true? I mean, if each cube $D_i$ is centered at some point lying at $C_k$ it is clearly true, but we don't know if this happens. I tried to give a proof that we can assume that each $D_i$ can be choosen in a way that is centered in $C_k$ but I couldn't prove that.
Can you help me to justify that assertion on the book? Thanks in advance.
I am reading "Analysis on Manifolds" by James R. Munkres.
Since $D_1,D_2,\dots$ actually intersect $E_k$, $D_i\cap E_k\neq\emptyset.$
Since $E_k=C_k\cap E$, $D_i\cap C_k\neq\emptyset.$
So, there exists $x_i\in\mathbb{R}^n$ such that $x_i\in D_i\cap C_k$.
Let $x$ be an arbitrary point of $D_i$.
Let $d$ be the center of $D_i$.
Then,
$|x-d|\leq (\text{ the width of }D_i)/2<\frac{\delta}{2}.$
$|x_i-d|\leq (\text{ the width of }D_i)/2<\frac{\delta}{2}.$
So, $|x-x_i|\leq |x-d|+|d-x_i|<\frac{\delta}{2}+\frac{\delta}{2}=\delta.$
Since $x_i$ is a point of $C_k$, $x$ is contained in the $\delta$-neighborhood of $C_k$.
$x$ was an aribitrary point of $D_i$.
So, $D_i\subset\text{ the }\delta\text{-neighborhood of }C_k\subset\operatorname{Int}C_{k+1}\subset C_{k+1}.$