Doubt in the use of induction in theorem 1.10 (Replacement theorem) in Friedberg's Linear Algebra I am having a hard time understanding the use of induction in the proof of Theorem 1.10 in Friedberg's Linear Algebra book.

The doubt that I have is that the author is applying induction on the size $m$ of any linearly independent set in the space and at last, concludes by induction that the theorem holds for all natural numbers $m$. But I don't get how is that possible since then we would have proved that for all linearly independent sets of arbitrary size $m$, $m \leq n$.
 A: I think I have resolved my doubt and since no one has yet answered, I am posting my answer. If someone finds some mistake, please correct me.
So here our induction hypothesis is the theorem itself. But since my doubt pertains only to proving the inequality, I will consider our induction hypothesis as follows:
$P(n):$ If $L$ is the linearly independent set with $m$ vectors, then $m \leq n$.
Now if $m \geq n$, then the 'then'-part in the induction hypothesis is $false$. Also following the regular proof as shown in the question, $m$ should be strictly less than $n\,$ i.e. $\,m \lt n$. Hence, our 'if'-part of the hypothesis is also $false$. Therefore, by the rules of propositional logic, our conditional statement $P(m)$ is TRUE $\forall \,m\,$ (since for $m \lt n$, $P(n)$ is $true$ by the given proof in the question).
In other words, the proof shown takes care of $m \geq n$ cases by contradicting the existence of linearly independent set of size $m$ since for that, $m$ should be strictly less than $n\,$ i.e. $\,m \lt n$ and the rest is taken care by the conditional nature of $P(n)$.
A: I think the original author's induction hypothesis as follows:

$P(m)$: there is a vector space spanned by $G$ with cardinality $n$. If a natural number $m$ is the cardinality of the linear independent set $L$, then $m \le n$.

In this hypothesis we only need to test all the natural numbers $m$ from $0$, and if it is the cardinality of a linearly independent set $L$, it must be less than or equal to $n$. After understanding this, the author's proof process is very clear.
In fact, I was reading this book, and I also got stuck here. I hope my understanding can help you.
