Def. Convergent Sequence
We say that a sequence $\{a_j\}$ in $\mathbb{R}^n$ converges to the limit $L∈\mathbb{R}^n$, if
$∀ε>0,∃J>0$ such that if $j≥J$ then $|a_j−L|<ε$.
(I think that might be a mathematical def. which based on FOL)
I'm trying to understand what this means, is it saying:
$$\forall\varepsilon>0,\exists J> 0,s.t.(j\ge J\rightarrow|a_j−L|<ε)$$
Update: $\color{lightgrey}{\text{(Thanks @Michael to point out, there are some mistakes in those steps)}}$
Steps:
apply $p\rightarrow q\Leftrightarrow \neg p\vee q$
$$\forall\varepsilon>0,\exists J> 0,(j<J\vee |a_j−L|<ε)$$
distribute the quantifiers $\color{lightgrey}{\text{(which is not valid)}}$
$$(\forall\varepsilon>0,\exists J> 0,s.t. j< J)\vee (\forall\varepsilon>0,\exists J> 0,s.t. |a_j−L|<ε)$$
Therefore
$$(\exists J> 0, s.t. j< J)\vee (\forall\varepsilon>0, |a_j−L|<ε)$$
equivalent to
$$\color{blue}{(\forall J> 0,j\ge J)}\rightarrow \color{orange}{\forall\varepsilon>0,|a_j−L|<ε}$$
$\underline{\text{What's the quantifier for $j$ here}}$
My thoughts
Since $j$ is an index, which we might have $j\in\mathbb{N}$, the definition might want to say when $j$ goes to $\infty$, something happens
I'll try to write the def. by myself first maybe, which might help to understand it, I guess it want to say something like
$$j\text{ approach } \infty\rightarrow a_j=L$$
$$\Leftrightarrow \color{blue}{j \text{ approach } \infty}\rightarrow \color{orange}{a_j-L=0}$$
I think the orange part are equivalent, so we have
$$\forall\varepsilon>0,|a_j−L|<ε\leftrightarrow a_j-L=0$$
The blue part might be equivalent, then we have
$$\forall J> 0,j\ge J\leftrightarrow j \text{ approach } \infty$$
If this is the case, the definition is somehow making sense
But I don't understand how is $(\forall J> 0,j\ge J\leftrightarrow j \text{ approach } \infty)$ hold
What the word approach really means, i know that $$\forall c\in\mathbb{R},c<\infty$$
And I can understand that
$$(\forall J> 0,j>J)\leftrightarrow j\equiv\infty$$
Since $\forall J>0, \infty\neq J$
So in the blue part, I think it's fine to replace $\ge$ with $>$, we have
$$\forall J>0,j> J\rightarrow a_j=L$$
$$\Leftrightarrow j\equiv \infty\rightarrow a_j=L$$
However, I personally feel there is some difference between, $j\equiv\infty$ and $j$ approach to $\infty$
And if this is the case, what I wrote above would make no sense.
Then I will stop for now $\dots$
Could someone expain this to me in details.
Thanks for your help.
Best Answer
We need an universal quantifier for $j$ :
$$∀ε>0 \ ∃J>0 \ ∀j \ (j≥J→|a_j −L|<ε).$$
The symbol $\lim_{n \to \infty}$ is a symbol that we cannot "split" into parts: there is no $\infty$ value such that $n=\infty$.
See e.g. Terence Tao, Analysis I (Springer, 3rd ed. 2016), page 129 :