1

There are sequences:
$\{x^n\}_{n\in N}$, where

$x^n=\langle x^n_1, x^n_2, x^n_3,...\rangle, n=1,2,...$
$x=\langle x_1,x_2,x_3,...\rangle$

How should I write that $x$ is limes of $x^n$? I use definition that
$lim_{n \rightarrow \infty}x_n=x$ iff $(\forall \epsilon >0)(\exists n_0\in N)(\forall n\geq n_0)(d(x_n,x)<\epsilon)$

(There are some properties of these sequences, but I think they are not important because I am confused just about how to write down expression)

I have to use $d_{\infty}(x^n,x)=sup\{|x^n_k-x_k|:k\in N\}$ metric.

I wrote it as
$(\forall \epsilon >0)(\exists p_0\in N)(\forall n\geq p_0)(\sup\{|x^n_k-x_k|:k\in N\}<\epsilon)$
but then I saw that teacher wrote it as
$(\forall \epsilon >0)(\exists p_0\in N)(\forall n\geq p_0)(\sup\{|x^k_n-x_n|:k\in N\}<\epsilon)$
and now I don't know what is correct.

user23709
  • 759
  • I think the confusion comes because you need first fix all inidices and define for what each are etc. Particularly $n$ and $k$. Then be consequent in waht you want to express. I think the formulation of your teacher if solely regarded what it says is the correct one, however he was not consequent with the indices and applies $n$ confusing when compared with the sequences. – al-Hwarizmi Jun 14 '13 at 16:45
  • I wrote that $x^n=\langle x^n_1,x^n_2,...,\rangle$, so k represents position of element in the sequence. And $x^n$ is just a mark for n-th sequence (since I have a lot of sequences, instead to write $x,y,z,...$, I write $x^1, x^2, x^3, ...$. I am confused about what should I fix - index of element in sequence and go through all sequences or I should fix a sequence and go through its elements. – user23709 Jun 14 '13 at 16:54

1 Answers1

2

I thinks this should make things clear what I mean. You may decide to write:

$$x_n=\langle x_{n,1}, x_{n,2}, x_{n,3},\dots,x_{n,k},\dots\rangle, n=1,2,...$$ $$x=\langle x_1,x_2,x_3,\dots,x_k,\dots\rangle$$ where $k$ represents position of element in the sequence. And $n$ is index for n-th sequence. $$\dots$$ $$(∀ϵ>0)(∃p_0∈N)(∀n≥p_0)(sup{|x_{n,k}−x_k|:k∈N}<ϵ)$$

PS: provided I understood all correct what you want to say.

al-Hwarizmi
  • 4,290
  • 2
  • 19
  • 36
  • If I understood, it is correct how I wrote, not how teacher wrote, right? – user23709 Jun 14 '13 at 17:16
  • I know what he wants to say. He is just liberal with his indices but says the same. He uses $k$ different. I would say both of you are not well formulated, suggest never to use exponent position for indexing. – al-Hwarizmi Jun 14 '13 at 17:28