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9 Cards in this Set
- Front
- Back
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Heine-Borel Theorem |
Any open cover of a closed, bounded interval always has a finite subcover. |
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Bolzano-Weierstrass Theorem |
Any bounded, infinite set of reals has an accumulation point |
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accumulation point |
A point x (in R) is an accumulation point of S if, for every e>0, there exists some s (in S) such that the distance between x and s is less than e. |
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open cover |
An open cover for a set S is a collection of open intervals {O_a : a is in A} such that each point in S is contained in at least one of these intervals. |
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subcover |
A sub cover for an open cover {O_a} consists of a collection of open intervals that together form an open cover for S. |
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function |
a function f is a collection of ordered pairs of elements such that if (a,b) and (a,c) are both in the collection, then b=c. |
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domain |
the domain of function f is the set of all x's such that (x,y) is in f for some y. |
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range |
the range of function f is the set of all y's such that (x,y) is in f for some x. |
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definition of a limit |
We call L the limit of f(x) as x approaches p if: For all e > 0, there exists some d > 0 such that if the (distance from x to p) < d then the (distance from f(x) to L) < e |
