How are the "limits" of infinite intersections and unions defined?


What can I use to infer what kind of set is spanned by e.g.

$$\bigcup_{i=1}^{\infty} [2 + 1/i, 5 - 1/i]$$

since to know this one would somehow have to infer the $\infty$ -limit of the above intersection, right?

  • $\begingroup$ There is no limit in the definition of such a union. It is just a shorthand for the union where $i$ ranges over all natural numbers. $\endgroup$ Nov 1 '16 at 13:18
  • $\begingroup$ @TobiasKildetoft I have read about there being no limit in the definition, but how does one infer the value at $\infty$ then? Or find the set that the above example spans? $\endgroup$
    – mavavilj
    Nov 1 '16 at 13:19
  • $\begingroup$ There is no value "at" $\infty$, as it is defined the way I said above (which does not include any $\infty$). $\endgroup$ Nov 1 '16 at 13:22
  • $\begingroup$ If $\mathcal{C}$ is a set, we define $\bigcup\mathcal{C}=\{x:\exists C\in\mathcal{C}$ s.t. $x\in C\}$. When $\mathcal{C}=\{C_i:i\in I\}$, we usually write $\bigcup\mathcal{C}=\bigcup_{i\in I}C_i$. Finally, when $I=\mathbb{N}$, it is usual to write $\bigcup\mathcal{C}=\bigcup_{i=1}^\infty C_i$. $\endgroup$ Nov 1 '16 at 13:23

The "limit" just is the union (or intersection). Another way, and IMO the less silly way, the above expression could be written is $\bigcup\{[2+1/i,5-1/i]:i\in \mathbb{N}\}$ (assuming $i$ ranges over $\mathbb{N}$). The notation used in your post is, I think, meant to somehow make the concept of arbitrary union or intersection familiar by analogy with the extension of ordinary addition to $\Sigma$-summation, but there's nothing deeper at work here than the notions of union/intersections themselves.


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