In Analysis 1 by Terence Tao and here and basically every proof I read, I find that the proof is done by induction.
But in the attached link the proof starts by stating that:
$\mid A\mid =n$ and $\mid A\mid =m$ implies the existence of the bijections $f:\mathbb{N_n}\rightarrow A$ and $g:\mathbb{N_m}\rightarrow A$. But since $g$ is bijective and hence invertible then $g^{-1}:A\rightarrow \mathbb{N_m}$ is also a bijection.
That implies that the composition $g^{-1}\circ f:\mathbb{N_n}\rightarrow \mathbb{N_m}$ is also a bijection.
After that the proof continues by induction.
What I'm asking is since there exists a bijection from $\mathbb{N_n}$ to $\mathbb{N_m}$ and hence by definition they have equal cardinality isn't that enough as a proof of unique cardinality? Why do we need induction?