In a proof by weak induction, we interpret $f$ to be a predicate statement depending on an integer $n \in \mathbb{Z}$ which we want to show to be true for all integers on some ray $[a, \infty) : = \{ n \in \mathbb{Z} : n \geq a \}$ where the statement being true for $n$ is understood to be encoded by $f(n) = 1$. To accomplish this, we must show that $\{ n \in \mathbb{Z} : f(n) = 1\} = [a, \infty)$.