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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. (6 points) Let g be a bounded continuous function on (0,1) and define f on (0,1) by f(x) = (1 - x2)g(x). Show f is uniformly continuous on (0,1). 3. (6 points) Let f be a function defined and continuous on all of R. Also let ao € IR and suppose (an) is the sequence defined by an = f(an-1) when 72 > 0 (so as = f(ao), a2 = (f(ao)), and so on). Suppose finally that the sequence (an) defined this way converges to a finite number L. Show that f(L) = L. 4. (6 points) Let (S, d) be the metric space S = R² with the usual Euclidean metric d. Let E be the set E = {(x,y) € S : 1 < x² + y2 < 2} Show E is connected. Hint: Try to show that E is path-connected. rove or disprove THREE of the following statements, for five points each (please note that ere are five options - the last two are on the next page): 5. (5 points) Let (gk) be a sequence of continuous functions on [0, 1] such that 2001 9k converges uniformly on [0, 1]. Set Mk = sup{(gk(x) : x € [0,1]}. Then K1 Mk converges. 6. (5 points) Let f be a continuous function on (0,1). Then f achieves at least one of its two possible extrema on its domain (i.e. either f achieves a minimal value, a maximal value or both). Note that weak minima and maxima count in this case (so a constant function is not a counterexample). 7. (5 points) Let f be a continuous function on [a, b] which is weakly increasing (recall that this means f f(x) < f(y) whenever X < y), and let xo € (a,b]. Then the left-handed limit lim f(x) exists.

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