Math 131 Fall 2018 103118 Introduction to lim sup and lim inf

Recall: Cauchy sequence; complete metric space. Theorem: Euclidean space is complete. Monotonic sequences. Monotonic Sequence Theorem. Definition of limit superior, limit inferior. Examples. Statement of characterization of lim sup. Comment: if lim sup equals lim sup, then sequence converges. Lemma: set of subsequential limits in a metric space is closed.