Limit point of Sets

is a limit point of if it is adhere to .

E.g.

  • , , is adherent to , so is a limit point
  • , , is not adherent to , so is a not limit point. We call an isolated point in this case.

Isolated point

is an isolated point of if but is not an adhere to of .

Remark of Limit Points

Can be used for L’Hospital’s rule when we need to remove point .

Limit point of Sequence

Relationship with Subspaces

The following two statements are equivalent: - is a limit point of sequence - There exists a subsequence of the original sequence which converges to

In Metric Space

Suppose is a sequence of points in metric space , and let , we say that is a limit point of iff for every and there exists an such that