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P1 Let PC 10, +00) be a set with the following property: For any k e Zso, there exists I E P such that kn s 1. Prove that inf P = 0. P2 Two real sequences {0,) and {0} are called adjacent if {a} is increasing. b) is decreasing, and limba - b) = 0. (a) Prove that, for all n, a, son, and deduce that both sequences are bounded. (b) Show that if {a} and {m} are adjacent, then they are both convergent and lim+ = lim,-toober P3 Let {n} be a sequence of positive real numbers and let L = lim supe Assume that L < 0. r .)". (a) Show that for every r > L there exists N such that an sr for every n N. (b) is part (a) true for r = L? If yes - prove it, if not - provide a counterexample. P4 Let A be a subset of the real numbers. Denote by Aº the set of interior points of A. by bd(A) the set of boundary points of A and el(A) the set of closed points of A. Prove that bd(A) = cl(A)\A”. (That is, the boundary of A is the closure of A with the interior points removed.) P5 Let S: [a, b] → [a, b] be a continuous function, where a, b are real numbers with a <b. Show that has a fixed point (.e., there exists a ela, 6 such that f(a) = x). P6 Let {n} be a sequence of real numbers and let L be a real number. Consider the two statements: (i) Ve > 0 3N: lan-L<€ VnN ; (ii) EN: Ve > 0 Jan - L<e Un > N. Show that (il) implies (i). Show by example that (i) does not necessarily imply (ii).

Answer 1

P5 let fi [a, b ] [a, b ] is continuous Define : g(a)= x-f(x). 7 g(a) = a-f(a) so because fala) € [a, b ] or f(a) = a. Also g (b) = b - f(b) 70 because f g (b) [a, b] or f(b) Sb. Then because I i continuous, by intermidiale value theorem there enish one root of g in [a, b ] » f how a fixed point in [a, b] le 7 at [a, b] with flo za