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: $\color{lightgrey}{\text{(Thanks @Michael to point out, there are some mistakes in those steps)}}$


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.

  • 3
    $\begingroup$ 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.) $\endgroup$ – spaceisdarkgreen Oct 10 '19 at 0:49
  • 1
    $\begingroup$ 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 $j=\infty$. $\endgroup$ – Michael Oct 10 '19 at 1:13
  • $\begingroup$ Agree, I changed the order of the quantifier which might be a mistake @Michael $\endgroup$ – Manx Oct 10 '19 at 1:27
  • $\begingroup$ 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. $\endgroup$ – SquishyRhode Oct 10 '19 at 2:43
  • $\begingroup$ I see, make sense @MauroALLEGRANZA $\endgroup$ – 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.

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