There is no smallest positive decimal number." proof by contradiction

Question

Suppose that the statement:  "There is no smallest positive decimal number." is false. Suppose the negation is true.
There IS a smallest positive decimal number.
That smallest positive decimal number is N.N > 0  [since N is a positive]
N < n [for every positive decimal number, since N is the greatest]
0 < N/2 < N N/2 is a positive decimal number and it is smaller then N.

This contradicts the supposition that N < n for every positive decimal number
N/2 is < N, so the supposition is false and the statement is true.

Does this qualify as a proof?

Answer 1

The statement "N > n [for every positive decimal number, since N is the greatest]" should have been " N < n [for every positive decimal number n, since N is the smallest]"

Answer 2

Your proof works, but needs some polished.

I would also suggest that, since you are talking about decimal numbers, it would be more intuitive to divide by $10$ rather than $2$.

Since $N$ is a positive decimal number, clearly $N/10$ is also a positive decimal number.