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The first question,

  1. $\forall k\in\mathbb{N}, \forall\epsilon>0, \exists\delta>0$,...

  2. $\exists k_0\in\mathbb{N}, \forall\epsilon>0, \exists\delta>0$,...

For 1, we know that $\delta$ depends on $k$ and $\epsilon$, we denote $\delta=\delta(\epsilon,k)$.

For 2, of course, $\delta$ depends on $\epsilon$, but $\delta$ depends on $k_0$? $\delta=\delta(\epsilon,k)$ or $\delta=\delta(\epsilon)$?

If $\delta=\delta(\epsilon,k)$ in 2, we have that 1$\rightarrow$2. If $\delta=\delta(\epsilon)$ in 2, 1 can't imply 2. I think that $\delta=\delta(\epsilon,k)$ in 2, but I'm not sure that I'm right.

The second question,

  1. $\forall\epsilon>0, \exists\delta>0,\exists k_0\in\mathbb{N}$,...

  2. $\exists k_0\in\mathbb{N}, \forall\epsilon>0, \exists\delta>0$,...

My question is that, can 4 imply 3?

The third question,

$\exists\epsilon_0>0$, $\exists\delta_0>0$, my question is that, $\delta_0$ must be correlated with $\epsilon_0$ (each other)?

Thank you in advance.

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With quantifiers of different types, the second one depends on the first: $$(\forall \epsilon > 0)(\exists \delta > 0) ...$$ $$(\exists N)(\forall n > N) ...$$

Quantifiers of the same kind that are next to each other can be rearranged: $$(\exists x)(\exists y)$$ is equivalent to $$(\exists y)(\exists x)$$

When there are several alternations of quantifiers, as in $$(\exists k_0)(\forall \epsilon > 0)(\exists \delta > 0)$$ then the final quantifier does depend on both of the previous ones. The rule about swapping quantifiers only applies when two quantifiers of the same kind are immediately adjacent.

In the two statements in the question, (1) does imply (2), for example if you take $k = 1$ and apply the first statement, this will prove the second statement.

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