Exercise 4.3.7

Proof.

(a)

The Tower Theorem shows that (a) and (b) are true for $m=2$. Suppose that (a) and (b) are true for an integer $m\ge 2$. We prove that they remain true for the integer $m+1$.

$\text{∙}$ If $[L_i:L_{i-1}]=\infty$ for some $i,1\le i\le m$, the induction hypothesis show that $[L_m:L_0]=\infty$. As $L_0\subset L_m\subset L_{m+1}$, the part (a) of Theorem 4.3.8 (Tower Theorem), shows that $[L_{m+1}:L_0]=\infty$.

Moreover, if $[L_{m+1}:L_m]=\infty$, this same part (a) of Tower Theorem gives also $[L_{m+1}:L_0]=\infty$.

For all $i,1\le i\le m+1$,

$$[L_i:L_{i-1}]=\infty \Rightarrow [L_{m+1}:L_0]=\infty ,$$

so the part (a) is proved for the integer $m+1$.

$\text{∙}$ Suppose that $[L_i:L_{i-1}]<\infty$ for all $i,1\le i\le m+1$. Then the induction hypothesis gives

$$[L_m:L_0]=\underset{1\le i\le m}{\prod }[L_i:L_{i-1}]$$

The part (b) of theorem 4.3.8 implies that

$$\begin{array}{llll}\hfill [L_{m+1}:L_0]& =[L_{m+1}:L_m]\times [L_m:L_0]& \hfill & \\ \hfill & =[L_{m+1}:L_m]\times \underset{1\le i\le m}{\prod }[L_i:L_{i-1}]& \hfill & \\ \hfill & =\underset{1\le i\le m+1}{\prod }[L_i:L_{i-1}].& \hfill & \end{array}$$

So the induction is done.