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1. (6 points) Let (fn) be the sequence of functions defined ...

1. (6 points) Let (fn) be the sequence of functions defined on [0,5] by for n = 1,2, Show directly from the definition (meaning you should not use any theorems we have proved about uniform convergence - theorems from earlier in the book are fine though) that (fn) converges uniformly on [0,5]. 2...

2. (a) (5 points) State the Mean Value Theorem. (b) (12 poi...

2. (a) (5 points) State the Mean Value Theorem. (b) (12 points) Suppose f : [0,3] R is continuous on [0,3] and differentiable on [0,3]. If 2 > f'(x) > 0 and f(0) = 4 then what is the largest value f(3) can be? 3. (a) (5 points) Suppose f : A R is a function. Define what it means fo...

(10 pts) Let E C R be a bounded set of reals. and f : E IR...

(10 pts) Let E C R be a bounded set of reals. and f : E IR a uniformly continuous function. Prove that f is bounded. Evaluate the integral So x+y(y - 2x)² dy)) dar using change of variables u = x + y. v = y - 2.x. Define f : R² R by 1 - cosey) x2 + y2 if y = 0 f(x,y) = otherwis...

Question 3 (a) (i) Let C be a subset of a metric space. Use...

Question 3 (a) (i) Let C be a subset of a metric space. Use sequences to state what it means to say that C is a compact set. (ii) Let C be a nonempty compact set in the metric space (IR, . I) and let p E R. Show that the set given by A={x+p|xEC}, = is compact. Note: It is of no importance whe...

Question 3: (b) Let (xn) be a sequence of real numbers such...

Question 3: (b) Let (xn) be a sequence of real numbers such that lim v In = 1. Is it true n->oo or false that in such case the series 00 In diverges? If it is true give a brief n=1 explanation and if it is false give a counterexample. Question 4: Determine whether the following series...

Question 1 (b) For points a = (a1, a2), b = (b1, b2) €...

Question 1 (b) For points a = (a1, a2), b = (b1, b2) € R², define = 1-611,122-1213. - (i) Prove from the definition that d is a metric on R². (ii) Show that the sequence ((x("),*()) nEN is convergent in (R², d) if the sequences (x(n) nEN and (x(n) ) nEN are converg...

Let on be a bounded sequence. In this workshop, we will deri...

Let on be a bounded sequence. In this workshop, we will derive several results about the numbers called lim inf an and lim supan- Let. us define new sequences bn and Cn as follows: In other words, bn is the greatest lower bound of the set {ax : kej and le n} and Gn is the least upper bound of the...

8 Let an 0 and [ an < 80. n=1 (a) Show that liminf...

8 Let an 0 and [ an < 80. n=1 (a) Show that liminf nan = 0. n->oo (b) Give an example showing that lim sup nan > 0 is possible. n->00 Let f : [a, b] R be continuous, and suppose that f takes on no value more than twice. Show that f takes on some value exactly once...

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