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Unformatted text preview: ………………………………………………………………………….” 4 4. (a) [1 mark] Formulate “The Monotone Sequence Theorem” (b) [5 marks] Show that exists and find its value if is defined k k x +∞ → lim } { k x recursively by 1 = x and 2 1 2 1 1 + = + k k x x 5 5. (a) [2 marks] Let denote the line segment in R i L 2 from the origin (0, 0) to the point ) 1 , 1 ( 2 i i , ( i = 1, 2, 3, …. .) on the curve . Is the union 2 ) ( x x f = S = compact? Justify your answer. i i L ∞ = 1 U (b) [4 marks] Suppose S ⊂ R n is compact, f : S R is continuous, and f ( x ) > 1 for → every x ∈ S . Show that there is a number c > 1 such that f ( x ) ≥ c for every x ∈ S . 6...
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 Fall '09
 RomauldStanczak
 Calculus, Topology, ........., li, General topology

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