I was reading Elementary Number Theory by David Burton and in the first chapter it had a proof of Principle of Finite Induction.
The principle of Finite Induction states that for a set $S$ containing positive integers such that:
a. $1 \in S$
b. If integer $k$ belongs to $S$, then $k+1$ also belongs to $S$.
Then $S$ is the set of all Positive Integers.
My proof was as follows:
Let $T$ be a set of positive integers that do not belong to S, such that $a$ is the least element of $T$. Then we have that $a\notin S$. So we must have that $a-1 \notin S$, because if it would have been in $S$, then by condition $(b)$, $a\in S$ which is a contradiction. So $a-1 \notin S \implies a-1 \in T$. This is a contradiction to assumption that $a$ is the least element. Thus $T$ is empty?
Is this a valid proof?
Actually I began in a similar way as David Burton, but some different paths..
Thank you! Please don't be angry because I have just started to learn these things!