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Real Analysis 1. Let X be any nonempty set. Suppose f: x - R is a bounded function on X and denote sup f = sup{f(x):x € x} and inf x Prove that - = x 2. Prove the following parts of the so-called "limit comparison theorem": Suppose Li-1 ak. LA-1 bs are both series with and that lim a L. bg (a) Prove that if o < L < 8 and 5.00 be converges, then LE-1 as also converges. (b) Prove that if L = 80 and EK-1 bg diverges, then as also diverges. 3. Suppose ! is defined and differentiable for every 3 > 0, and f(x) - o as x DO. Set g(x) = f(=+1) - f(z). Prove that 9(2) 0 as 2 DO. 4. Suppose I : ja, b] - R is Riemann integrable. Using the result from problem #1, show that f is also a Riemann integrable function by proving that for any E> 0 there exists a partition P such that You may not apply the theorem which states that the composition of a continuous function with an integrable function is integrable. 5. Let R = [a.b x |c.d] be a rectangle in R². (a) A function P: R R is said to have separated variables if N P(2.y) k-1 for some scalars Ck € R and functions 9 continuous on ja.t and |c.d] respectively. Prove that if h(2,g) is continuous on R, there exists a sequence Pn of functions with separated variables such that P, h uniformly on R as nz - DC. (b) Use the previous part to show the following elementary version of Fubini's theorem: If h is continuous on R, then da dy. 6. Let E CR" be an open set and suppose f:E R is differentiable on its domain. Prove that if f has a local maximum at a point I E E. then Df(x) = o. 7. Let f - R" be of class C (all partial derivatives exist and are continuous); suppose that f(a) = o and that Df(a) has rank n. Show that if cis a point of R" sufficiently close to 0, then the equation f(2) = e has a solution. 8. Given a.b> 0, let E be the region bounded by the ellipse S+ = 1, that is, E = Show that the area of E is sab in two ways: (a) By computing HE dA with a change of variables. (b) By Green's theorem.

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