117-022609 - MATH 117 LECTURE NOTES DR JULIE ROWLETT 1...

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MATH 117 LECTURE NOTES FEBRUARY 26, 2009 DR. JULIE ROWLETT 1. Finally... Functions! Analysis is the study of limits. So far, we’ve learned a lot about limits including (1) Topology: accumulation points are like limits for metric spaces. Closed sets contain all their accumulation points, so this is one way you have seen limits in the context of topology. (2) Real numbers: the real numbers have the least upper bound property: every set which has an upper bound has a supremum (least upper bound). This makes them complete: a metric space is complete if every Cauchy sequence has a limit. (3) Sequences: conditions which guarantee a sequence converges, methods for computing the limit of a convergent sequence. (4) Subsequences, liminfs and limsups. What do the subsequences tell you about the original sequence? All the work you have done with limits should give you a strong foundation for your future analysis studies. In particular, you are now prepared to consider limits of functions. Continuity, derivatives, integrals: these are all defined by limits. So,
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