What is wrong with representing an arbitrary natural and odd square number as $(2n-1)^2$?

Question

Could I not represent every odd square number in $\mathbb N$ using the following notation: $(2n-1)^2$ where $n \in \mathbb N$. For every $n= 1,2,3...$ I get the set $1,9,25...$ every odd square number. Why can't I do this?

I ask because then I could prove that if a natural numbers square is odd then the number itself is odd using a direct proof (which I am told cannot be done) as follows:

$(2n-1)^2= k^2$, where $k \in \mathbb N^{odd}$

$\Longrightarrow$ $\sqrt{(2n-1)^2}= \sqrt{k^2}$

$\Longrightarrow$ $(2n-1)= k$

Obviously I am starting from a wrong assumption in mind. Thanks in advance.

Answer 1

What if there is an even number which squares to an odd number? This is actually the key to the proof that you can represent any odd square in this way. Your 'proof' would be circular.

Answer 2

Better show the fact : If the square of a number is odd, than the number is odd.

indirectly :

It is easy to see that the square of an even number is even. Conclusion : If the square of a number is odd, then the number is odd, because if it were even, the square would be even.

If you have this fact, it is immediately clear that the odd squares can be represented by $(2n-1)^2$ because every odd number can be represented by $2n-1$.

This way, you can avoid the circulation shown by Daniel Littlewood.