Order of quantifiers and the intepretation of a sequence $(a_n)$.

Question

I've been struggling for the past few days with a problem related to the interpretation of a predicate when the order of the quantifiers is changed.

Predicate. A sequence of real numbers $(a_n)$ is convergent if,

$$ \exists L \in \mathbb R: \forall \epsilon>0: \exists n_0 \in \mathbb N: \forall n \in \mathbb N: n>n_0 \Rightarrow \mid a_n - L\mid <\epsilon. $$

Assume the order of the quantifiers are changed to obtain the following predicates;

$$ \text{(a) } \exists L \in \mathbb R: \exists n_0 \in \mathbb N: \forall \epsilon>0: \forall n \in \mathbb N: n>n_0 \Rightarrow \mid a_n - L\mid <\epsilon, $$

$$ \text{(b) } \exists L \in \mathbb R: \forall \epsilon>0: \forall n \in \mathbb N:\exists n_0 \in \mathbb N: n>n_0 \Rightarrow \mid a_n - L\mid <\epsilon, $$

$$ \text{(c) } \forall \epsilon>0: \exists L \in \mathbb R: \exists n_0 \in \mathbb N: \forall n \in \mathbb N: n>n_0 \Rightarrow \mid a_n - L\mid <\epsilon. $$

How do these changes affect the (interpretation/meaning of) sequence $(a_n)$, and what type of sequences satisfy to these predicates. More importantly, I would also appreciate some explanation about how to approach these type of problems.

Your many answers/feebacks/tips are greatly appreciated.

Answer 1

1. This implies the sequence converges to a limit $L$ and also constant (equal to $L$) from some $n_0$.
2. This implies the sequence has some sub-sequence that converges to some limit $L$.
3. This is equivalent to saying A sequence of real numbers ($a_n$) is convergent if... (normal definition).

Answer 2

(a) This means that the sequence $(a_n)$ is constant from some $n_0$ onwards.

(b) This statement is true for any sequence of reals $(a_n)$. In fact it is true in a "trivial" kind of way, since for any given $n$ we can always pick an $n_0$ such that $n_0 \geq n$. (For concreteness, we can simply pick $n_0 = n$.)

(c) One can show that this statement is equivalent to the original definition; i.e., for $(a_n)$ a given sequence of reals, it is equivalent to the statement "$(a_n)$ is a convergent sequence". However, it is important to note that the proof of the non-trivial implication of this equivalence requires the use of the completeness of the real numbers (if instead we were talking about sequences of rational numbers, for instance, the implication wouldn't be true). The proof isn't very difficult yet also not trivial.