I have a question regarding Theorem 3-14 in Spivak's Calculus on Manifolds:
The proof starts by considering a closed rectangle $U\subset A$ such that all the sides of $U$ are of the same length $l$. Let $\epsilon>0$. Spivak claims that if $N$ is sufficiently large and $U$ is divided into $N^n$ rectangles, with sides of length $l/N$, then for each of these rectangles $S$, if $x\in S$ we have
$$|Dg(x)(y-x)-(g(y)-g(x))|<\epsilon|x-y|\leq \epsilon\sqrt{n}(l/N)$$
for all $y\in S$. I don't get why this is necessarily true. My attempt to prove this is as follows. Take any $\epsilon>0$. Since $g$ is differentiable on $U$, for all $x\in U$ there exists $\delta'_x >0$ such that for all $y$,
$$|x-y|<\delta'_x\implies|Dg(x)(y-x)-(g(y)-g(x))|<\epsilon|x-y|.$$
Since $g$ is continuously differentiable, for all $x\in U$ there exists $\delta''_x$ such that for all $y$ and $v$,
$$|x-y|<\delta''_x\implies|Dg(x)(v)-Dg(y)(v)|<\epsilon|v|.$$
Let $\delta_x=\min\{\delta'_x,\, \delta''_x\}$. Then $\{B(x;\delta_x/2)\,|\,x\in U\}$ is an open cover of $U$. Since $U$ is compact, a finite number of these balls cover $U$. Say these balls are centered around $x_1,\,x_2,\,\ldots,\,x_k$. Let $\delta=\min\{\delta_{x_1}/2,\,\delta_{x_2}/2\,\ldots,\,\delta_{x_k}/2\}$. Pick any arbitrary $x\in U$. Then $x\in B(x_i,\delta_{x_i}/2)$ for some $1\leq i \leq k$. Thus
$$|Dg(x_i)(x-x_i)-(g(x)-g(x_i))|<\epsilon|x_i-x|.$$
For any $y\in B(x;\delta)$ we have $y\in B(x_i,\delta_{x_i})$, ergo
$$|Dg(x_i)(y-x_i)-(g(y)-g(x_i))|<\epsilon|x_i-y|.$$
Combining these two statements gives
$$|Dg(x_i)(x-y)-(g(x)-g(y))|<\epsilon(|x_i-x|+|x_i-y|).$$
Again using the fact that $x\in B(x_i,\delta_{x_i}/2)$, we have
$$|Dg(x_i)(x-y)-Dg(x)(x-y)|<\epsilon|x-y|,$$
hence
$$|Dg(x)(y-x)-(g(y)-g(x))|<\epsilon(|x-y|+|x_i-x|+|x_i-y|).$$
Unfortunately, I don't know how to progress from here. Any help would very much be appreciated.
Best Answer
Hint : You cannot show what you want like this : the $(x_i)$ introduced prevents you to have a bound with $|x-y|$. The reason is you must show that the inequality :
$|Dg(x)(y-x) - (g (y) - g (x)) | < \epsilon |x - y|$
is true for both $x$ and $y$ variyng (not only for a $x $ fixed and a $y $ varying). You need the continuity of $Dg$ to show this. Then you use the compacity of $U \times U $ to conclude.