Could anyone verify my proof for strong induction?
Proof: Let $S$ be a set of positive integers containing some integer $a$. Suppose $S$ has the property that $n$ belongs to $S$ whenever every integer less than $n$ and greater than or equal to $a$ belongs to $S$. As $S$ contains $a$ it will also contain $a+1$ and ,as a result, also $a+2$ and so on. For the sake of contradiction, let us assume $S$ does not contain all positive integers. Let $T$ be the set of integers not contained in $S$. By the Well-Ordering Principle, $T$ must have a least value, $b$, as it will be non-empty. But then, this means $b-1$ is in $S$ meaning by the condition on $S$, $b$ must be in the set. This is a contradiction so $S$ must contain all positive integers.