An ABC soft question about epsilon-delta argument Someone told me that some textbooks present epsilon-delta argument somewhat misleadingly. For example, consider the simplest one: the convergence of the sequence $(1/n)_{1}^{\infty}$
to $0$.
These textbooks may prove this convergence as follows: Let $\epsilon > 0$. Since $n \geq N,$ we have $1/n \leq 1/N,$ so that if we choose $N := [1/\epsilon] + 1$ (where [x] denotes the greatest integer not greater than the given number $x$) then $n \geq N$ implies $1/n < \epsilon$.
Yes, such proof looks to prove the implication $n \geq N \implies 1/n < \epsilon.$ But indeed this implication is assumed valid and what requires to prove is the existence of $N$ for every $\epsilon > 0.$ 
I do not know how to reply to such question, would anyone please help? 

" ...what requires to prove is the existence of N for every ϵ>0."
The Archimedean property of real numbers ensures that for every real $\epsilon > 0$ there exists a positive integer $N$ such that
$$N > \frac1{\epsilon}$$
