Editing Set-theoretic limit (section)

==Examples==
* Let <math>A_n = \left(- \tfrac{1}{n}, 1 - \tfrac{1}{n}\right].</math> Then
<math display=block>\liminf_{n \to \infty} A_n = \bigcup_n \bigcap_{j \geq n} \left(-\tfrac{1}{j}, 1 - \tfrac{1}{j} \right] = \bigcup_n \left[0, 1 - \tfrac{1}{n}\right] = [0, 1)</math>
and
<math display=block>\limsup_{n \to \infty} A_n = \bigcap_n \bigcup_{j \geq n}\left(-\tfrac{1}{j}, 1 - \tfrac{1}{j}\right] = \bigcap_n \left(- \tfrac{1}{n}, 1\right) = [0, 1)</math>
so <math>\lim_{n \to \infty} A_n = [0, 1)</math> exists.
* Change the previous example to <math>A_n = \left(\tfrac{(-1)^n}{n}, 1 - \tfrac{(-1)^n}{n}\right].</math> Then
<math display=block>\liminf_{n \to \infty} A_n = \bigcup_n \bigcap_{j \geq n} \left(\tfrac{(-1)^j}{j}, 1-\tfrac{(-1)^j}{j}\right] = \bigcup_n \left(\tfrac{1}{2n}, 1 - \tfrac{1}{2n}\right] = (0, 1)</math>
and
<math display=block>\limsup_{n \to \infty} A_n = \bigcap_n \bigcup_{j \geq n} \left(\tfrac{(-1)^j}{j}, 1 - \tfrac{(-1)^j}{j}\right] = \bigcap_n \left(-\tfrac{1}{2n-1}, 1 + \tfrac{1}{2n-1}\right] = [0, 1],</math>
so <math>\lim_{n \to \infty} A_n</math> does not exist, despite the fact that the left and right endpoints of the [[Interval (mathematics)|intervals]] converge to 0 and 1, respectively.
* Let <math>A_n = \left\{ 0, \tfrac{1}{n}, \tfrac{2}{n}, \ldots, \tfrac{n - 1}{n}, 1\right\}.</math> Then
<math display=block>\bigcup_{j \geq n} A_j = \Q\cap[0,1]</math>
is the set of all [[rational number]]s between 0 and 1 (inclusive), since even for <math>j < n</math> and <math>0 \leq k \leq j,</math> <math>\tfrac{k}{j} = \tfrac{nk}{nj}</math> is an element of the above. Therefore,
<math display=block>\limsup_{n \to \infty} A_n = \Q \cap [0, 1].</math>
On the other hand, <math display=block>\bigcap_{j \geq n} A_j = \{0, 1\},</math> which implies
<math display=block>\liminf_{n \to \infty} A_n = \{0,1\}.</math>
In this case, the sequence <math>A_1, A_2, \ldots</math> does not have a limit. Note that <math>\lim_{n \to \infty} A_n</math> is not the set of accumulation points, which would be the entire interval <math>[0, 1]</math> (according to the usual [[Euclidean distance|Euclidean metric]]).