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4. (20 pts) Let {xn} be a Cauchy sequence. Show that a) (5 pts) {xn} is...
5a) (5 pts) Find lim inf (xn) and lim sup (rn), for rn = 4 + (-1)" (1 - 2). Justify your answer 5b) (5 pts) Find a sequence r, with lim sup (xn) = 3 and lim inf (x,) = -2. 5c) (10 pts) Let {x,} be a bounded sequence of real numbers with lim inf (x,) = x and lim sup (x,) = y where , yER. Show that {xn} has subsequences {an} and {bn}, such that an...
13 14 Exercise 13: Let (xn) be a bounded sequence a S be the set of limit points of (n), i.e. S:{xER there exists a subsequence () s.t. lim } ko0 Show lim inf inf S n-o0 Hint: See lecture for proof lim sup Exercise 14: (Caesaro revisited) Let (x) be a convergent sequence. Let (yn) be the sequence given by Yn= n for all n E N. Show that lim sup y lim sup n n-+00 n o0
Let (xn) be a bounded sequence of real numbers, and put u = lim supn→∞ xn . Let E be the set consisting of the limits of all convergent subsequences of (xn). Show that u ∈ E and that u = sup(E). Formulate and prove a similar result for lim infn→∞ xn . Thank you! 7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...
#s 2, 3, 6 2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for all n in N, show that linnAn = 0 if and only if lim-= oo. 3. Let ()nEN be a sequence in R. (a) If x <0 for all n in N, show that - -oo if and only if xl 0o. (b) Show, by example, that if kal → oo,...
Exercise 2.3.7: Let {xn} and {yn} be bounded sequences. a) Show that {Xn+yn} is bounded. b) Show that (lim inf xn) + (lim inf yn) < lim inf (Xn tyn). noo Hint: Find a subsequence {Xn; +yn;} of {Xn +yn} that converges. Then find a subsequence {Xnm;} of {Xn;} that converges. Then apply what you know about limits. n->00 c) Find an explicit {{n} and {yn} such that noo (lim inf xn) + (lim inf yn) <lim inf (Xn+yn). noo...
ANSWER 1 & 2 please. Show work for my understanding and upvote. THANK YOU!! Problem 1. Let {x,n} and {yn} be two sequences of real numbers such that xn < Yn for all n E N are both convergent, then lim,,-t00 Xn < lim2+0 Yn (a) (2 pts) Prove that if {xn} and {yn} Hint: Apply the conclusion of Prob 3 (a) from HW3 on the sequence {yn - X'n}. are not necessarily convergent we still have: n+0 Yn and...
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above, x = lim sup (r) if and only if For all 0 there is an NEN, such that x <x+e whenevern > N, and b. For all >0 and all M, there is n > M with x - e< In a. Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above,...
ANSWER 5,6 & 7 please. Show work for my understanding and upvote. THANK YOU!! Problem 5. (3 pts) Let {x,n} be a bounded sequence of real numbers and let E = {xn : n E N}. Prove that lim inf,,0 In and lim inf, Yn are both in E. Hint: Use the sequential characterization of the closure, i.e., Proposition 3.2 from class. Problem 6. (3 pts) As usual let Q denote the set of all rational numbers. Prove that R....
(6) Let (2,A, /i) be a measure space. Let fn: N -» R* be a sequence of measurable functions. Let g, h : 2 -> R* be a integrable pair of measurable functions such that both are on a set AE A and g(x) < fn(x) < h(x), for all x E A and n e N. Prove that / / fn du lim sup fn d lim sup lim inf fn d< lim inf fn du п00 n oo...
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and y1 = 8 and for n = 1,2,3,··· x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y . nn nn (a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all positive integers n. (xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive integers n. Hence, prove...