Ex3-28

Chapter 3, Question 28 
Question: Prove the following assertion that is made in the proof of Theorem 3.33: If T:\mathbb R^n\to \mathbb R^n is a Lipschitz transformation, then there is a constant c'>0 such that |TI|\leq c'|I| for every interval I. 
Proof: First we prove when I=Q is a bounded cube in \mathbb R^n. Let d be the side length of Q, then \text{diam} Q=\sqrt{\sum_{k=1}^nd^2 }=\sqrt nd. Let c be the Lipschitz constant of T, so that |Tx-Ty|\leq c|x-y|\quad\text{ for all }x, y\in \mathbb R^n. 
Fixed x\in Q , then for any y\in Q we have
|Tx-Ty|\leq c|x-y|\leq c\cdot\text{diam }Q 
Thus TQ is contained in an n - dimensional sphere centered at Tx with radius c\cdot\text{diam }Q . Therefore, if we let Q' be a cube with side length 2c\cdot\text{diam }Q , centered at Tx , we will have TQ\subset Q' , hence
|TQ|\leq |Q'|=(2c\cdot \text{diam }Q)^n=(2c)^n n^{n/2} d^n =c' |Q| 
Where c'=(2c)^n n^{n/2} . Note that this constant c'is independent of the choice of cube, even if |Q|=+\infty the inequality is still true, no matter |TQ| is finite or not. Thus
|TQ|\leq c'|Q|\quad\text{ for each cube }Q\subset \mathbb R^n.
This next part isn't necessary, since the book defines "interval" as "closed interval". But I suppose there is no harm in leaving it up. 
For a general interval I\subset \mathbb R^n, given a positive \varepsilon we can choose an interval J such that I\subset J^\circand |J|\leq (1+\varepsilon)|I|. Using Theorem 1.11, which states that every open set in \mathbb R^n can be written as a union of nonoverlapping closed cubes, we can write J^\circ = \bigcup_{k=1}^\infty I_k such that each I_k is a closed cube; by Corollary 3.24 we can further write |J^\circ| = |J| = \sum_{k=1}^\infty |I_k|. Thus, 
\begin{align}|TI|&\leq |T(J^\circ)|_e\\
&=\left|T\left(\bigcup_{k=1}^\infty I_k\right)\right|_e\\
&=\left|\bigcup_{k=1}^\infty TI_k\right|_e\\
&\leq \sum_{k=1}^\infty|TI_k|\\ 
&\leq c'\sum_{k=1}^\infty |I_k|\\
&=c'|J|\\
&\leq c'(1+\varepsilon)|I|.\end{align}Since \varepsilonis arbitrary, we conclude |TI|\leq c'|I| for every interval I\subset \mathbb R^n.