# Confusion regarding Cauchy's General Principle and Uniform Convergence

The definitions of the two are so alike, that it confuses me.

Cauchy's General Principle:

The necessary and sufficient condition that a function $$f(x)$$ may tend to a definite limit, say $$l$$, as $$x \to a$$, is that:

If $$\forall \epsilon>0$$, $$\exists$$ a $$\ \delta>0$$ such that $$|f(x_1)-f(x_2)| < \epsilon$$ whenever $$0<|x_1-a|<\delta, \ 0<|x_2-a|<\delta$$

Uniform Continuity:

A function $$f$$ defined on a domain $$D$$ is said to be uniformly continuous on the set $$S$$ $$( S \subset D)$$ if:

$$\forall \ \epsilon >0$$, $$\exists$$ a $$\delta>0$$, such that $$|f(x_1)-f(x_2)|< \epsilon$$ for any two points $$x_1, x_2 \in S$$ with $$|x_1-x_2| < \delta$$.

First of all, I understand that the former definition is a "Local Concept" i.e. it focuses around the particular point $$a$$ , while the other one is a "Global Concept", i.e. defined all over the set $$S$$. Another thing is, the sign "$$0<$$ " in the General Principle. It deletes the point "$$a$$" from the neighbourhood of both $$x_1$$ and $$x_2$$, while the other does not. But the similarity is, both definitions give independence to choose arbitrary pair of points within a certain neighbourhood.

Is anything wrong with my perception? Am I missing anything very obvious?

Please do provide further insight.

simple convergence : given a $$\in$$ I, $$\forall \epsilon>0, \exists \delta$$ such as $$|x-a|<\delta \implies |f(x)-f(a)|<\epsilon$$