I've been reading the book "Real analysis" by Royden and I have a major doubt in the way the book make use of the induction argument. In the Proposition 6 he state the following:
Let A be any set and $\{E_{k}\}_{k = 1}^{n}$ a finite disjoint collection of measurable sets, then
$$m^*\bigg(A \cap \bigg[\bigcup\limits_{k=1}^{n} E_{k}\bigg]\bigg) = \sum_{k = 1}^n m^*(A \cap E_k)$$
and the proof start like this:
The proof proceeds by induction on n. It is clearly true for $n = 1$. Assume it is true for $n - 1$. Since the collection...
I omitted the part of the proof that I have no problem with. Meaning that I have no problem with the proof itself except for the lines I've invoked. Here is my humble doubt:
Why is it correct to start with the base case $n = 1$? Because as far as I know the idea of using induction is that the previous step helps to do the next step, perfectly seen when in the commom last part of induction proofs the case $n = k - 1$, helps to derive an argument for the case $n = k$.
To be more precise, what Royden's book is saying (to me) is that the case n = 1, meaning $m*(A \cap E_1) = m*(A \cap E_1)$ (which, by the way, is utterly true) helps to prove the case $n = 2$, meaning $m*(A \cap (E_1 \cup E_2)) = m*(A \cap E_1) + m*(A \cap E_2)$. But I cannot see how the base case, $n = 1$, helps to develop an argument for the case $n = 2$. Of course the case $n = 2$ can be derive from the context (definition of $m^*$ and its properties) but not for the base case $n = 1$.
It seems to me that the base case should be the case $n = 2$. Because from this case we can derive an argument for the case $n = 3$ and from this, an argument for the case $n = 4$ and so on.
But maybe I am wrong (otherwise Royden and many others proofs I've seen would have a flaw in their arguments), so please help me to work this out. Is it correct the way I understand induction arguments? meaning that in induction proofs the previous step must help to do the next step, even in the very first or base case. Thanks in advance.