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In definitions and exercises, I notice that "so that" and "such that" are seemingly used interchangeably. Are they in fact interchangeable, or is one more appropriate for a specific context?

Note: $\mathrm{Dom}\,(f)$ means the domain of $f$.

Example 1:

Suppose that a function $f$ is continuous at a point $c$ and $f(c) > 0$. Prove that there is a $\delta > 0$ $\color{red}{\text{so that}}$ for all $x \in \mathrm{Dom}\,(f)$, $ |x-c| \le \delta \ \Rightarrow \ f(x) \ge \frac{f(c)}{2} $

Example 2:

A function $f(x)$ is continuous at a point $c \in \mathrm{Dom}\,(f)$ if and only if for each $\varepsilon > 0$ there is a $\delta > 0$ $\color{red}{\text{such that}}$ for all $x \in \mathrm{Dom}\,(f)$: $ |x - c| \le \delta \ \Rightarrow \ |f(x) - f(c)| \le \varepsilon $

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    Dear @Andrew: In my view there is no ambiguity: formulas _always_ bind more tightly than text, so the "for all $n$" can only apply to all of "$a(n)\neq0$". Assuming that, as Milne suggests, one could take the negation bar out of the formula and make it say "NOT ($a(n)=0$ for all $n$)" is what I consider ridiculous. Note that one can't even do this is you pronounce "$\neq$" as "is unequal to".2012-08-16

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Seeing as no one has posted their response as an answer, I simply don't want this to come up under "Unanswered Questions." For the actual answer to this question, consider the comments.

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    Sorry about that, I left my computer and hadn't come back to accept it yet because when I posted that it said I had to wait 45 minutes2012-08-18