What is the proof that induction is a valid technique?
It makes perfect intuitive sense to me. If I know that $P(n_0)$ is true for some natural number $n_0$ and I know that $P(k)$ implies $P(k+1)$ for all natural numbers $k$, then $P(n_0)$ being true implies that $P(n_0+1)$ is true, which implies that $P(n_0+2)$ is true, which implies that $P(n_0+3)$ is true, which implies... And then I ultimately deduce by observing this pattern that $P(n_0+j)$ is true for all natural numbers $j$.
However, that doesn't actually seem rigorous. How do I know that the pattern continues indefinitely? Is there some fundamental axiom I'm just missing? What is it?