MATH 210 Exam 2, Spring 2013

a : Oe American University of Beirut Hib Math 210 29 aC 46 UU lé ; Exam 2, April 6, 2013 oe ¢ 1. (20 points) Using the definition 0 n+3nt! en f the limit of a sequence. show that 1 oe no y +ntl 2 tim no /4 +] et OS), é ” sar all neW . define 2. (10 points) Suppose that a sequence fa, , converges to 4. Foran —— jo. le — Fat Int , e 2 Using the definition of the limit of a sequence, show that {b, ‘" _ converges to a 3. (20 points) a) State the definitions of “a subsequence of a given sequence” , “a C auchy sequence in a metric space”; and “a complete metric space”. oO . ’ f »= : ~ RBI Fe OMe A ort b) Let {a,}'_, be a convergent sequence in R. Show that 14, ;,_,15 Cauchy and bounded 4. (15 points) Suppose 4 is nonnegative for each k €/N . Show that 4 ] j a a ig 7 < a 2 : a, OR: converges iff = converges. kel tal 1+, 5. (20 points) a) Using the definition, show that ) log( ae to - log — fl ee m 2 b) Suppose da, is a convergent series of positive terms. n=] oo oo Are the series es and V4, necessarily convergent? Justify you four answe n=] r answer (15 points) Let :(0,1) ->R be defined by f(x)=+. x 6. A a) Using the definition ofa continuous function, show that f is continuous on (0 1). b) Show that / is not uniformly continuous on (0,1),