notes / Math 55b / chapter 1: Metric and Topological Spaces / From my notebook
N3 Sequences and Limits
Definition N3.1 (Limit). (as ) converges to a limit (written , or ) iff for all there exists such that for all , .
Definition N3.2 (Cauchy sequence). A sequence in a metric space is a Cauchy sequence if for all there exists such that for all , .
Question N3.3. If a sequence converges then it is Cauchy, but not necessarily vice versa. Why?
Definition N3.4 (Complete). is complete if every Cauchy sequence converges.
Example N3.5. is complete; with the usual distance is not complete.
Example N3.6. : show that the partial sums
are Cauchy.
is the completion of (all the limits of sequences that are Cauchy).
Proposition N3.7. in a metric space iff for every open there is such that for all , .
Closed sets
Definition N3.8 (Closed). is closed if is open.
Note. Most subsets are neither closed nor open. Closed not open.
Proposition N3.9. If a set is closed, then given any sequence in which converges to , we have . (?)