Advanced Calculus: Difference between revisions
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Advanced Calculus as taught in [https://bookstore.ams.org/amstext-5/ Fitzpatrick's book]. | Advanced Calculus as taught in [https://bookstore.ams.org/amstext-5/ Fitzpatrick's book]. | ||
This is content covered in MATH410 and MATH411 at | This is content covered in MATH410 and MATH411 at UMD. | ||
==Sequences== | ==Sequences== | ||
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===Closed=== | ===Closed=== | ||
The folllowing definitions of Closed Sets are equivalent. | The folllowing definitions of Closed Sets are equivalent. | ||
* (Order) | * (Order) A set <math>S</math> is open if for every number <math>x \in S</math>, there is an epsilon <math>\epsilon > 0</math> such that the epsilon ball is a subset of s: <math>\{ y | d(x, \epsilon) \} \subset S</math>. The compliment of an open set is closed. | ||
* (Sequences) A set <math>S</math> is | * (Sequences) A set <math>S</math> is closed if it contains all its limit points. That is <math>\forall \{x_i\} \subseteq S</math>, <math>\{x_i\} \rightarrow x_0 \implies x_0 \in S</math>. | ||
; Notes | ; Notes | ||