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I am learning this wiki page, which uses sequence of sets in the definition

Suppose that ${\displaystyle \{A_{n}\}_{n=1}^{\infty }}$ is a sequence of sets. The two equivalent definitions are as follows.

Using union and intersection, define

$\liminf_{n \rightarrow \infty} A_n = \bigcup_{n \ge 1} \bigcap_{j \geq n} A_j$

...

The sequence ${A_n}$ is said to be nondecreasing if each $A_n ⊂ A_{n+1}$

the simplest example of a (monotonic increasing) sequence I can imagine is the Natural number $(0, 1, 2, …)$

is there a concrete example in this context of could be used to illustrate the nondecreasing sequence of sets?

for instance, is this sequence $(\{0\}, \{0, 1\}, ..., \{0, 1, 2, …\})$ an nondecreasing sequence of sets (I am not sure it is, I use this as a inspiration)?

JJJohn
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    Your sequence is in fact a strictly increasing sequence of sets. Consider the sequence of sets ${A_n}$, where $A_n$ is the set of primes $\le n (n\ge 2)$. So the sets are $A_2={2},\ A_3={2,3},A_4=A_3,A_5={2,3,5}$, and so on. This sequence is non-decreasing. – Samrat Mukhopadhyay Aug 01 '19 at 12:31
  • @SamratMukhopadhyay thanks for your comments. Is $A_1={2}$ in your example – JJJohn Aug 01 '19 at 12:39
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    @yaojp: He defined the sequence only for $n\ge 2$, but applying the definition to $n=1$ would give $A_1=\varnothing$ because there are no primies $\le 1$. – hmakholm left over Monica Aug 01 '19 at 13:48

2 Answers2

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The sequence $\{\{0\}, \{0, 1\}, ..., \{0, 1, 2, …\}\}$ is indeed an increasing sequence of sets.

Another example is $A_n=[1,n]$, i.e. $\{\{1\}, [1,2], [1,3], \cdots [1,n] \cdots\}$

Note that $[1,2]$ for example is an interval containing all real numbers between $1$ and $2$, but if you want only the natural numbers, then $[1,2]=\{1,2\}$

Also $A_n=]-n,n[$ is an ascending chain of sets of real numbers as the following intervals $\{]-1,1[, ]-2,2[, \cdots , ]-n,n[, \cdots \}$ but also you can pick up (if thats what you need) only the natural numbers from these intervals, and you'll get $]-2,2[=\{0,1\}$ and $]-3,3[=\{0,1,2\}$, and that will be an ascending chain of sets of natural numbers.

Note that the notation $]-n,n[$ is written in some references as $(-n,n)$

A third example is taking the sets of even numbers for examples which means $\{\{0\}, \{0,2\}, \{0,2,4\}, \cdots \{0,2,4,6, \cdots \}\}$

I hope that this answers your question.

Fareed Abi Farraj
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To make sure that you understand that an non-decreasing sequence of sets is completely unrelated to whether the elements of that set increase, let me show you another strictly increasing sequence of sets: $$\{100\},\{2,100\},\{2,7,100\},\{2,7,100,4711\},\{2,7,42,100,4711\},\ldots$$ As you see, in each step, something is added and nothing is removed, but it doesn't matter whether what is added is larger or smaller than what was added in the previous steps.

Note also that it doesn't matter how many items you add in each step, thus for example, the following sequence is also non-decreasing: $$\{1\},\{1,2\},\{1,2,3,4\},\{1,2,3,4,5,6,7,8\},\ldots$$

Finally, while all the previous examples are strictly increasing sequences, non-decreasing just means nothing is removed, therefore the following is also a non-decreasing sequence of sets: $$\{1\},\{1\},\{1\},\{1\},\{1\},\ldots$$

celtschk
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  • Thanks for your answer. ${100},{2,100},{2,7,100},{2,7,100,4711},{2,7,42,100,4711},\ldots$ is a clear example and what is the formula of this sequence? – JJJohn Aug 01 '19 at 21:43
  • @yaojp: I didn't have a formula in mind for that sequence. The rule is simply “in each step, add a new number”. There probably is some formula that reproduces those terms I've given and satisfies the rule. But it doesn't matter, and searching for such a formula would be a waste of time. There is no need that a sequence is given by a closed formula. – celtschk Aug 01 '19 at 23:20