Thread: The definition of the limit

1. The definition of the limit

Hey guys, we have started not long ago to learn the term of limes.

So the known definition of the limit of a series goes like that:

If
$a_n$ converges to 0, then $\forall \epsilon>0$, there exists an $N \in \mathbb{N}$ s.t. $n \ge N$ $\implies$ $|a_n - 0|<\epsilon$

I have some elementary questions about the definition, would it be correct if I use the definition a little bit? such that:

* If $\forall N \in \mathbb{N}$ $\exists \epsilon >0$ s.t. $\forall n >N$ we get $|a_n - L|<\epsilon$ $\implies$ $\lim_{n \to \infty}a_n = L$

* If there exists $N \in \mathbb{N}$ s.t. $\forall \epsilon>0$ $\forall n >N$ we get $|a_n - L|<\epsilon$ $\implies$ $\lim_{n \to \infty}a_n = L$

And another type of limes question:

If the series $b_n$ is defined by $b_n = |a_n - L|$ and converges to 0 then $\lim_{n \to \infty}a_n = L$ ?

Thank you!

2. Re: The definition of the limit

Originally Posted by CStudent
Hey guys, we have started not long ago to learn the term of limes.

So the known definition of the limit of a series goes like that:

If
$a_n$ converges to 0, then $\forall \epsilon>0$, there exists an $N \in \mathbb{N}$ s.t. $n \ge N$ $\implies$ $|a_n - 0|<\epsilon$

I have some elementary questions about the definition, would it be correct if I use the definition a little bit? such that:

* If $\forall N \in \mathbb{N}$ $\exists \epsilon >0$ s.t. $\forall n >N$ we get $|a_n - L|<\epsilon$ $\implies$ $\lim_{n \to \infty}a_n = L$

* If there exists $N \in \mathbb{N}$ s.t. $\forall \epsilon>0$ $\forall n >N$ we get $|a_n - L|<\epsilon$ $\implies$ $\lim_{n \to \infty}a_n = L$

And another type of limes question:

If the series $b_n$ is defined by $b_n = |a_n - L|$ and converges to 0 then $\lim_{n \to \infty}a_n = L$ ?
If $\forall N \in \mathbb{N}$ $\exists \epsilon >0$ s.t. $\forall n >N$ we get $|a_n - L|<\epsilon$ $\implies$ $\lim_{n \to \infty}a_n = L$

Look you must start with Suppose that $C>0$. Everything in the proof depends upon that positive number.
The expression $|L-a_n|$ is the distance from $a_n$ to $L$.
So expression $|L-a_n|<C$ means that is the distance from $a_n$ to $L$ is less than $C$.
Now $|L-a_n|<C$ is the open interval $(L-C,L+C)$ so that $a_n\in(L-C,L+C)$.
This also means that you can find a positive integer $N_1$ (it depends upon $C$) having the property that picking any integer, $j$, greater than $N_1$ then $a_j\in(L-C,L+C)$ or $|a_j-L|<C$

So roughly speaking all that means is Given a positive error, C, we can find a place in the sequence where from there on the terms are close to L.

3. Re: The definition of the limit

Originally Posted by Plato
If $\forall N \in \mathbb{N}$ $\exists \epsilon >0$ s.t. $\forall n >N$ we get $|a_n - L|<\epsilon$ $\implies$ $\lim_{n \to \infty}a_n = L$

Look you must start with Suppose that $C>0$. Everything in the proof depends upon that positive number.
The expression $|L-a_n|$ is the distance from $a_n$ to $L$.
So expression $|L-a_n|<C$ means that is the distance from $a_n$ to $L$ is less than $C$.
Now $|L-a_n|<C$ is the open interval $(L-C,L+C)$ so that $a_n\in(L-C,L+C)$.
This also means that you can find a positive integer $N_1$ (it depends upon $C$) having the property that picking any integer, $j$, greater than $N_1$ then $a_j\in(L-C,L+C)$ or $|a_j-L|<C$

So roughly speaking all that means is Given a positive error, C, we can find a place in the sequence where from there on the terms are close to L.
Great explanation, thank you!

4. Re: The definition of the limit

Am I the only one who can't see the equations? All I'm seeing is a bunch of CodeCogs icons saying "equation quota exceeded."

-Dan

5. Re: The definition of the limit

Dan, you are not the only one who does not see the equation. I see the same thing as you do.

6. Re: The definition of the limit

Originally Posted by topsquark
Am I the only one who can't see the equations? All I'm seeing is a bunch of CodeCogs icons saying "equation quota exceeded."
-Dan
Try to reply with quote. I see the code at first, but then after replying I see the mathematics.

7. Re: The definition of the limit

How do I fix it?

8. Re: The definition of the limit

Originally Posted by CStudent
How do I fix it?
Replace $[\text{tex}]$ or $[\text{math}]$ tags with \$\$$'s? 9. Re: The definition of the limit Originally Posted by SlipEternal Replace [\text{tex}] or [\text{math}] tags with \$$'s?
It's just like MathJax?

10. Re: The definition of the limit

Originally Posted by CStudent
It's just like MathJax?
Yup.

-Dan