In the following, let : [0, ) [0, ) be defined by f(x) = x^. (a)...

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In the following, let : [0, ) [0, ) be defined by f(x) = x^. (a) Show that f is strictly increasing for all n = N, and use it to conclude that f is injective. We say f is strictly increasing if f(x) < (y) for all x, y = [0, ) with x < y. (b) Show that f is continuous for all n N. Then, given M N, use part (a) and what you know about continuous functions to show that the restriction f| 10,M] : [0, M] [0, M] is both surjective and injective, and hence bijective. (c) Use the results of part (b) to conclude that for any a [0, ), there exists a unique non-negative x such that x = a. In the following, let : [0, ) [0, ) be defined by f(x) = x^. (a) Show that f is strictly increasing for all n = N, and use it to conclude that f is injective. We say f is strictly increasing if f(x) < (y) for all x, y = [0, ) with x < y. (b) Show that f is continuous for all n N. Then, given M N, use part (a) and what you know about continuous functions to show that the restriction f| 10,M] : [0, M] [0, M] is both surjective and injective, and hence bijective. (c) Use the results of part (b) to conclude that for any a [0, ), there exists a unique non-negative x such that x = a.