Can someone clarify whether this can be said as a proof?

Question

A person claims this to be a demonstration and not a proof my question is ,does it fit the bill of being a valid Mathematical proof?

Answer 1

I disagree with the statement that “This, however, is not possible as $n$ is unbounded above.” However, in order to prove that, given $\varepsilon>0$, there is some $n\in\mathbb N$ such that $10^{-n}<\varepsilon$, you can use the Archimedian property: there is some $n\in\mathbb N$ such that $n>\frac1\varepsilon$ and therefore$$10^{-n}=\frac1{10^n}<\frac1n<\varepsilon.$$