Proving there is no smallest positive number

[Uranian]

1. The problem statement, all variables and given/known data
"True or false: there is a smallest positive number. Explain."


2. Relevant equations
N/A, but for practice I'll try my hand at phrasing it mathematically:
\forallx\in(0,∞)\existsz\in(0,∞):(z<x)


3. The attempt at a solution
My issue with the question is mathematically proving it - I'm a bit paranoid because I've been losing a lot of marks on communication and I don't think it'll be enough for me in this particular class to simply say that the statement is false because there is an infinite amount of numbers between 0 and 1. So, I was thinking it could be proven in a way similar to how we prove there is no largest real number...
Let z be the smallest positive real number such that 0<z<x where x\in(0,∞):
let x=z-1
then:
z<z-1
0<-1 which is not true. Therefore, the statement is false and there is no smallest positive number.
Is this a logical argument? This is my first course in proofs, and I'm a freshman, so I don't feel very confident in constructing my arguments. Mainly I would just like some feedback, and if I'm doing something wrong, could someone hint towards the correct argument...? Any response is much appreciated : )

[ArcanaNoir]

If z<x, why does x=z-1 ? I would try a contradiction. Let x= the smallest positive number. Then there is no number z such that x>z>0. Let z=x/2.... its a little course in the phrasing but you see what I'm trying to do?

[Uranian]

Right, that is a much better argument...I suppose I just misunderstood the proof that there is no largest real number which I came across in my calculus text. : \

[ArcanaNoir]

I do that all the time. Flip a sign here, switch all for exists there, and before you know it, you're proving the wrong thing. It got me once on a test >_<

[SammyS]

I see that you apparently understand the problem now.

One way to approach it would be to ask yourself, if given a positive number, x, what number is between x and zero?