In a proof by weak induction on the natural numbers, we interpret $f$ to be a predicate statement depending on a natural number $n \in \mathbb{N}$ which we want to show to be true for all natural numbers $\mathbb{N}$ where the statement being true for $n$ is understood to be encoded by $f(n) = 1$. To accomplish this, we must show that $\{ n \in \mathbb{N} : f(n) = 1\} = \mathbb{N}$.