Let : R R for n N be functions such that and g as n uniformly on the set E R, where f, g : R R are functions. Show that as n uniformly on the set E.
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For each n N let : [a,b] R where a,b R are such that a < b. Show that if F : [a,b] R is such that F as n in a pointwise fashion for each n N the function is monotonically increasing, then F is monotonically increasing.
Fix , There exist positive integers , such that for all , and for all , . Define . Then for all , which proves it.
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