# Can I do this instead to prove the Strong Principle of Induction (Tao 2.2.14)?

I have already read the following (this one and this one too) discussions on Stack Exchange and they have not answered my query. Proposition 2.2.14 asks the reader to prove that:

Proposition $$2.2.14$$ (Strong principle of induction). Let $$m_0$$ be a natural number, and let $$P(m)$$ be a property pertaining to an arbitrary natural number $$m$$. Suppose that for each $$m \ge m_0$$, we have the following implication: if $$P(m')$$ is true for all natural numbers $$m_0 \leq m' < m$$, then $$P(m)$$ is also true. (In particular, this means that $$P(m_0)$$ is true since in this case, the hypothesis is vacuous .) Then we can conclude that $$P(m)$$ is true for all natural numbers $$m\geq m_0.$$ (Hint: define $$Q(n)$$ to be the property that $$P(m)$$ is true for all $$m_0≤m note that $$Q(n)$$ is vacuously true when $$n)

Instead of what Tao suggests, can I let $$Q(n)$$ to be the property that $$P(m)$$ is true for all $$m_0\le m?

Yes. That would be saying exactly the same thing. Picking $$m_0+n$$ as the upper bound is the same as picking $$n'=m_0+n$$ as the upper bound like Tao suggested. Just a different way of writing it.