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

discrete mathematicspredicate-logicquantifiers

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.

Best Answer

  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).
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