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Unformatted text preview: 4130 HOMEWORK QS Due never Solutions to these problems will not be supplied ( but see one of us if you want to know how to do a particular problem). These problems are not necessarily representative of what will be on the exam, but they are left over from a list of problems which I did not have time to put into any of the homeworks or exams. Another good source of problems is anything which you were asked to prove as an exercise in the lecture notes! Also, there are plenty of problems in the textbook. 1. Standard problems (1) Let { x n } be a convergent sequence and k 0. Show that { x n + k } also converges. (2) Show that an unbounded monotone sequence either diverges to or diverges to . (3) Show that if a sequence { x n } has a finite liminf and limsup, then it is a bounded sequence. (4) Give an example of open sets U,V such that U 6 = V but U Q = V Q . (5) Using the intermediate value theorem, show that the continuous image of an interval is an interval. (6) Show that f ( x ) = x is differentiable on (0 , ). (7) Suppose f : ( a,b ) R is a continuous bounded strictly increasing function. Show that the image of f is the interval ( c,d ) where c = inf x ( a,b ) f ( x ) and d = sup x ( a,b ) f ( x ). (8) Let f : A R be continuous, where A R . Suppose x A and f ( x ) &lt; 0. Show that there is a neighbourhood U of x and a natural number N N such that f ( y )  1 /N for all y U . (9) Consider the power series f ( x ) = X p P x p 1 where P = { 2 , 3 , 5 , 7 ,... } denotes the set of prime numbers. What is the radius of convergence of f ( x )? Does the series f (1) converge? Does the series f ( 1) converge?...
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This note was uploaded on 09/22/2010 for the course MATH 413 at Cornell University (Engineering School).
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