(adsbygoogle = window.adsbygoogle || []).push({}); 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:

[itex]\forall[/itex]x[itex]\in[/itex](0,∞)[itex]\exists[/itex]z[itex]\in[/itex](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[itex]\in[/itex](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 : )

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# Homework Help: Proving there is no smallest positive number

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