1. Suppose C1, C2, C3, ... is a nondecreasing sequence of sets, i.e., Ck C Ck+1, for k= 1, 2, 3, . . . . Then find lim k→∞ Ck, if Ck = {(x, y) : 1/k≤ x² + y² ≤ 4 − 1/k}, k = 1, 2, 3, .... Justify your answer.
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- Define the sequence {Sn} by Sn= [ 1 / (2)(1) ] + [ 1 / (3)(2) ] + ............. + [ 1 / (n+1)(n) ] Prove that Limn→∞ Sn = 1.Let {xn} and {yn} be sequences such that lim yn = 0. Suppose that for all k ∈ ℕ an for all m ≥ k we have |xm - xk| ≤ yk. Show that {xn} is Cauchy.If Ean, converges and a, > 0 for all n, can anything be said about E(1/an).?
- 6) What is the limit of T(n) as n -> ∞?In a bag of 345 chocolate candies, 36 of them are brown. The candy company claims that 13% of its plain chocolate candies are brown. For the following, assume that the claim of 13% is true, and assume that a sample consists of 345 chocolate candies. Complete parts (a) through (e) a. For the 345 chocolate candies, use the range rule of thumb to identify the limits separating numbers of brown chocolate candies that are significantly low and those that are significantly high.Suppose that we observe that X1, X2, . . . , Xn are iid∼ U(0, 1). Show that X(1)converges in probability to zero.
- 1. Use the definition of the limit ( epsolon - delta ) to show thatlim of 1/z as z approaches -i2. Give the condition which ensure that |ez| < 1 where z in C.In a bag of 405 chocolate candies, 43 of them are brown. The candy company claims that 13% of its plain chocolate candies are brown. For the following, assume that the claim of 13% is true, and assume that a sample consists of 405 chocolate candies. Complete parts (a) through (e) below. a.) For the 405 chocolate candies, use the range rule of thumb to identify the limits separating numbers of brown chocolate candies that are significantly low and those that are significantly high. Values of ____ brown candies or fewer are significantly low. Values of ____ brown candies or greater are significantly high. Based on the results, is the result of 43 brown chocolate candies significantly low? Why or why not? b.) Find the probability of exactly 43 brown chocolate candies. c.) Find the probability of 43 or fewer brown chocolate candies.Prove using the ϵ−n0 definition that the sequence Xn=(9−7n)/(8−13n) converges, and find its limit.
- If {A_c} up ∞ down c=1is a convergent sequence , then {1/A_c} up ∞ down c=1 is a divergent sequence. T/FLet X be a set, and define d(x,y) ={0, if x=y; 1, otherwise Show that, if a sequence (xn) is convergent under the discrete metric, then (xn) must be eventually constant.3. If g : A → R is continuous and (an) is a Cauchy sequence in A,does it follow that the sequence (g(an)) is Cauchy?