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What is the need of induction in proving uniqueness of cardinality of sets?

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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?


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