MATH1833 Midterm: MATH 1833 UNB Exam 1833 99
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Read this first: please read each question carefully. Use the axiom of completeness to prove that an increasing sequence of real numbers that is bounded above converges. Prove that lim an = 0 if and only if lim |an| = 0. State the - de nition of what it means for a function f : r r to be continuous at c r. Suppose that f : r r is continuous at c and that the sequence (xn) converges to c. Use the - de nition of continuity to prove that the sequence (cid:16)f (xn)(cid:17) converges to f (c). Final exam: let f : r r and let c r. (a) Let f : [0, 1] r be given by. [10] f (x) =(0 x = 0 x2 sin 1 x x (0, 1]. Suppose that f : r r such that for all x r, f (x) = 0.
