Please help understand proof of small span theorem in multivariable calculus

Question

The proof of small span theorem (given in Apostol Volume 2) is as follows:

Please (if possible) give a detailed proof of this theorem including all information in the above proof

$\mathrm{off}$ is to be read as "of $f$"

Answer 1

We can pass this argument in $\mathbb R$. In $\mathbb R^n$ the interval is to be thought of as $$[a,b] = \{\lambda a+(1-\lambda)b \mid 0\leq \lambda\leq 1 \}. $$ The central part to the argument concerns that infinite set of subintervals.

Assuming, for a contradiction, the theorem is false, after every bisection, the finite set of subintervals (as a partition) does not satisfy the condition for $f$. More precisely, every finite partition contains at least one subinterval whose span of $f$ is at least $\varepsilon _0$.

Start generating finite partitions by bisection. Firstly $[a,b] = [a,x_1]\cup [x_1,b]$. By hypothesis, at least one of them has span of $f$ at least $\varepsilon _0$. Suppose it's $[a,x_1]$. Bisect again by $[a,x_1] = [a,x_2]\cup [x_2,x_1]$ and pick a subinterval whose span of $f$ is at least $\varepsilon _0$. Proceed analogously and obtain an infinite set of subintervals $I_k$, where $I_{k+1}\subseteq I_k$ for every $k\in\mathbb N$.

Put $t := \sup \left\{ \min\limits_{u\in I_k} u \mid k\in\mathbb N \right\}$. Now use continuity of $f$ at $t$ and the fact that the subintervals $I_k$ get smaller.

---

For the sake of clarity, I understand span of $f$ of some interval $I$ to be the diameter of its image under $f$ i.e $\mathrm{span}_f(I) := \sup \{\|f(x)-f(y)\| \mid x,y\in I\} = \mathrm{diam}(f(I))$. I've never heard it being called the span, though.