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F is strictly increasing at each point in (a,b)
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[QUOTE="mathwonk, post: 6053501, member: 13785"] did they prove the heine borel property in the section on lub's? (i.e. every open covering of a closed bounded interval by open intyervals has a finite sub covering.) given points c< d in the interval, the fact that f(c) < f(d) seems to follow from the existence of a finite cover of the closed bounded interval [c,d] by open intervals of the type mentioned. or did they discuss greatest lower bounds? given c in (a,b) you want to show there are no points x >c where f(x) ≤ f(c). Assuming there are some, the set of such x is a bounded non empty set and has a greatest lower bound d. That means every x with f(x) ≤ f(c) has x ≥ d, and also that there are such x arbitrarily close to d and greater than d. case 1) f(d) ≤ f(c). rule this out. case 2) f(d) > f(c). Rule this out. [/QUOTE]
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F is strictly increasing at each point in (a,b)
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