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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?

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    $\begingroup$ I think these textbooks are trying to lead you through how to get to the solution rather than set out a formal proof. When I was taught epsilon-delta proofs we were told to figure out a suitable $N$ in rough and then write out the argument formally $\endgroup$
    – JC574
    Aug 14 '14 at 15:38
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" ...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

$$N > \frac1{\epsilon}$$

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I think these textbooks are trying to lead you through how to get to the solution rather than set out a formal proof.

When I was taught epsilon-delta proofs we were told to figure out a suitable N in rough and then write out the argument formally.

So If we were told to prove $1/n \rightarrow 0$ as $n \rightarrow \infty$ then in rough I would think about how $n \ge N \implies 1/n \le 1/N $ and how I can use it to prove the limit.

Then I would state what we're trying to show:

$$\forall \epsilon > 0 \ \ \exists N \in \mathbb{N} \ \ n \ge N \implies 1/n < \epsilon $$

and then as my proof I would write -

"pick $N \in \mathbb{N}^{>0} $ such that $1/N<\epsilon . \ \ \ $ Then $\forall n \ge N \ \ \ $ $1/n \le 1/N < \epsilon $"

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  • $\begingroup$ Thank you for your efforts. But my question is how to convince somebody that does not accept this! $\endgroup$
    – Megadeth
    Aug 14 '14 at 15:49
  • $\begingroup$ they do not accept what in particular? If their claim is that textbooks present these proofs misleadingly, then I'd have to say I agree. $\endgroup$
    – JC574
    Aug 14 '14 at 15:52

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