https://math.unm.edu/~crisp/courses/math401/tao.pdf
Page 25, Lemma 2.2.3
The proof is done by showing Induction on n, $\textbf{keeping m fixed}$.
This means that this proof is valid for only that particular value of m.
But the Lemma claims for any natural number m.
This means that something is missing in my logic.
So does that fact that there was nothing special in the choice of m prove that it holds for any natural number m? But how do we write that Rigorously??