Equivalent definition of limit of a function (Reference request) Let $f: \mathbb R \to \mathbb R$, $x_0 \in \mathbb R$. We write 
$$\lim_{x\to x_0} f(x) =  L$$
if for all $\epsilon>0$, there is $\delta >0$ so that 
$$|f(x) - L | <\epsilon$$
whenever $|x - x_0|<\delta$, $x\neq x_0$. In Rudin's PMA it is shown that 
$$\lim_{x\to x_0} f(x) = L$$
if and only if 
$$\lim_{n\to \infty} f(x_n) = L$$
for all sequence $x_n \to x_0$, $x_n \neq x_0$. 
One can easily extend this to: 
$f$ has a limit at $x_0$ if and only if $\{f(x_n)\}$ is Cauchy whenever $x_n \to x_0$, $x_n \neq x_0$. 
Using this again, one can show that:
$f$ has a limit at $x_0$ if and only if: $\forall \epsilon>0$, there is $\delta >0$ such that $|f(x) - f(y)| < \delta$ whenever $x, y\in (x_0 - \delta, x_0 + \delta)\setminus \{x_0\}$. 
(Sketch of proof): $(\Rightarrow)$ is obvious. For $(\Leftarrow)$, let $x_n \to x_0$, then it is easy to check that $\{f(x_n)\}$ is Cauchy. Thus limit exists.
This has the adventage that one does not need to specify $L$ (the limit) in the definition. However, I cannot find this in Rudin's PMA. Has anyone seen this somewhere else or is there any mistake in my argument?
 A: Your argument is correct. I think the reason it is not used is because it is closer to "Cauchy" than "has a limit", and so is not equivalent in non-complete metric spaces. For example, the function $f$ defined on $\Bbb Q\setminus\{0\}\to\Bbb Q$ by $f(x)=\frac{F_{n+1}}{F_n}$ for $\pm x\in[\frac1n,\frac1{n+1})$ has limit $f(x)\to\varphi$ as $x\to 0$ in the usual sense on the reals, but as a function on $\Bbb Q$ it does not have a limit even though it is "Cauchy" at $0$ in the sense you have defined.
A: Yes, "your" statement was very popular in classic Italian texbooks. Now I do not have them on my desk, but I believe that your theorem appears more or less explicitly in Prodi's book "Analisi matematica" and in the old treatise by Luigi Amerio.
In Rudin's book it is not stated because Rudin approaches every topic from a rather abstract point of view. Limits are introduced for functions between metric spaces, where Cauchy sequences need not converge at all. Let me say that Rudin's attitude is always towards generalization as opposed to particular cases. Restricting the definition of limits to complete metric spaces would be rather useless, since limits are defined in topological spaces as well.
