Questions about basic logic (why position of "for all" makes difference)

Question

I am reading Appendix B of "Introduction to Analysis by Arthur Mattuck 1st edition" It says that the following two sentences have different meaning.

The book says that if the epsilon is introduced in the first clause, everything coming later is understood to depend to it. It says that N depends on the epsilon in the first sentence but does not depend on it in the second sentence. However, I don't understand what the difference is. Could you explain me difference? Also, this is my first question on stack exchange. If you think my attitude is not acceptible or doesn't follow the rules of stack exchange, please let me know so that I can fix it. Thank you

Answer 1

Hint

why position of “for all” makes difference ?

Consider a simple example with natural numbers and their ordering, i.e. the structure $(\mathbb N, <)$.

Can you see the difference between :

$\forall n \exists m \ (n < m)$

and :

$\exists m \forall n \ (n < m)$ ?