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the sequence notation is confusing me and my questions

usually, a pair of curly brackets {} could be used to denote a set of elements.

for instance, the set of Natural number that contains number 0 could be denoted by

${\displaystyle \;\{0,1,2,...\}=\mathbb {N} _{0}={\mathbb {N} }\cup \{0\}}$

this part seems to be a common agreement.

How about sequence?

this post and this post are using a pair of parentheses to denote a sequence, things like

(2, 3, 5, 7, 11, 13, 17, ...)

although another notation also has a number of upvotes, I'll go with the parentheses notation.

that's why I use this notation

$(1, 2, … , n)$

to denote a sequence of numbers.

so, a sequence of sets could be denoted by this notation

$(\{1\},\{1,2\},...,\{1,2,…,n\})$

all above is preparation for the final part

Suppose that ${\displaystyle \{A_{n}\}_{n=1}^{\infty }}$ is a sequence of sets.

is it clear and safe to write down this?

$\{A_{n}\}_{n=1}^{\infty} = (\{1\},\{1,2\},...,\{1,2,…,n\})$

which is equivalent to

$A_1 = \{1\}, A_2 = \{1,2\}, ..., A_n = \{1,2,…,n\}$

should I put ... at the end?

$(\{1\},\{1,2\},…,\{1,2,…,n\},\ldots)$

JJJohn
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    That is not clear and safe. The confusion is caused by you using curly braces to denote the actual sequence - ${A_{n}}{n=1}^\infty$ - when it would make much more sense to write $(A{n}){n=1}^\infty$. You can make this explicitly clear by writing also the sequence as $({x\in\mathbb{N}:x\le n}){n=1}^\infty$. – Peter Foreman Aug 01 '19 at 16:05
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    You should definitely put dots at the end, as otherwise you would have a finite sequence, also known as tuple. That is, $(1,2,3,4,5,6,7)$ has exactly $7$ members, as has $(1,2,3,\ldots,7)$. – celtschk Aug 01 '19 at 23:27

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Firstly, you should definitely put dots at the end to denote that is a infinite sequence, so you got

$(\{1\},\{1,2\},...,\{1,2,…,n\},...)$

Secondly, you've already chosen the parentheses notation

$\{A_{n}\}_{n=1}^\infty$ could be is this form $(A_{n})_{n=1}^\infty$

And then, put them together, you got

$(A_{n})_{n=1}^\infty = (\{1\},\{1,2\},...,\{1,2,…,n\},...)$

Lastly, you could write this sequence as

$(A_{n})_{n=1}^\infty = (\{x\in\mathbb{N}:x\le n\})_{n=1}^\infty$

JJJohn
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