definition of a number approach to $\infty$ in first-order-logic

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

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.

• The way to write it formally would be $$(\forall \epsilon>0)(\exists J\in \mathbb N)(\forall j\ge J)(|a_j-L|<\epsilon)$$ (Still this isn't quite in bare first order logic since we're using abbreviations in "bounding" the quantifiers. For instance $(\forall \epsilon>0) X$ is an abbreviation for $\forall \epsilon(\epsilon>0\to X)$ But it quickly becomes hard to read.) – spaceisdarkgreen Oct 10 '19 at 0:49
• This first thing is the definition: $$\forall\varepsilon>0,\exists J> 0,s.t.(j\ge J\rightarrow|a_j−L|<ε)$$ This second thing is certainly not equivalent: $$\color{blue}{(\forall J> 0,j\ge J)}\rightarrow \color{orange}{\forall\varepsilon>0,|a_j−L|<ε}$$ why are you trying to move the parentheses? If $|a_j-L|<\epsilon$ for all $\epsilon>0$ then $a_j=L$. And stand-alone $(\forall J >0, j \geq J)$ seems to mean that you have $j=\infty$. – Michael Oct 10 '19 at 1:13
• Agree, I changed the order of the quantifier which might be a mistake @Michael – Manx Oct 10 '19 at 1:27
• It does seem pertinent to specify what sets the variables come from. J and j are naturals. epsilon is a real. The sequence a_j and the limit L dont have to be reals, or even numbers, as long as a suitable metric defined by the absolute values is chosen. But that might be getting a little too topological. – SquishyRhode Oct 10 '19 at 2:43
• I see, make sense @MauroALLEGRANZA – Manx Oct 10 '19 at 7:31

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 $$|a_j−L|<ε$$.

What's the quantifier for $$j$$ here ?

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 :

We sometimes use the phrase “$$a_n \to x$$ as $$n \to \infty$$” as an alternate way of writing the statement “$$(a_n)^{\infty}_{n=m}$$ converges to $$x$$”. Bear in mind, though, that the individual statements $$a_n \to x$$ and $$n \to \infty$$ do not have any rigorous meaning; this phrase is just a convention, though of course a very suggestive one.