# Using induction for a specific set of natural numbers

I'm trying to understand how to use induction to prove a claim not for every natural number $$n$$, but only for specific natural numbers, say $$0 \leq n \leq k$$ for some positive $$k$$.

After proving $$P(0)$$, does it make sense to assume the proposition is true for $$P(n)$$ when $$0 \leq n \leq k - 1$$? In this way, I won't have to consider $$n = k + 1$$, and I believe the proof follows for all such $$n$$.

• Yes, you can use strong induction, but you need to show the induction step nevertheless, namely $P(k)$. Commented Dec 31, 2023 at 9:59

Let the property $$Q$$ be defined as $$Q(n)=P(n)$$ for $$n\leq k$$ and $$Q(n)=\text{true}$$ for $$n> k$$. Then showing that $$Q$$ is true for all $$n$$ is equivalent to show that $$P(n)$$ is true for all $$n\leq k$$. Now proving $$Q(n)$$ to be true for all $$n$$ by induction obviously means that
1. $$P(0)$$ has to be true and
2. P(n+1) is true provided that $$P(n)$$ is true for all $$n\leq k-1$$.