Problem 3: Let (w).>o be a sequence such that bn is convergent. Let (an)nzo to be...
Exercise 2.3.9. (a) Let (an) be a bounded (not necessarily convergent) sequence, and assume lim bn = 0. Show that lim(anon) = 0. Why are we not allowed to use the Algebraic Limit Theorem to prove this?
Let (an)nen be a bounded sequence in R. For all n e N define bn = sup{am, On+1, On+2,...}. (You do not have to show that the supremum exists.) (a) Prove that the sequence (bn)nen is a monotone sequence. (b) Prove that the sequence (bn)nen is convergent. (c) Prove or disprove: lim an = lim bre. 100 000
Problem 3. Fix any p > 1. Let ai, ..., An and bi, ..., bn be real num- bers. Prove that 1/ /n 1/P > ja, + 6P ( * wr)" (£wr)" (Mr) P + 1 (Hint: Minkowski inequality for a proper probability space.)
all three questions please. thank you Prove that for all n N, O <In < 1. Prove by induction that for all n EN, ER EQ. Prove that in} is convergent and find its limit l. The goal of this exercise is to prove that [0, 1] nQ is not closed. Let In} be a recursive sequence defined by In+1 = -) for n > 1, and x = 1. Prove that for all ne N, 0 <In < 1....
Let ao 2 bo > 0, and consider the sequences an and bn defined by an + bn n20 (1) Compute an+l-bn+1 1n terms of Van-v/bn. (2) Prove that the sequence an is nonincreasing, that the sequence bn Is nonde- creasing, and that an 2 bn for all n 20 (3) Prove that VanVbn S Cr for all n20, where C> 0 and y>1 (give values of C and γ for which this inequality holds). Conclude that an-bn C,γ-n, where...
Problem 3. Prove that if bn + B and B < 0, there is an N E N such that for all n > N, bn < B/2.
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...
theorem1 let an and bn be squences of real numbers theorem 2 let an and bn and cn be squences of real numbers if an<bn<cn theorem 3 let an be squences of real numbers if an=L and L defined at all an,f(an)=f(L) theorem 4 f(x) defined for all x>n0 then limit f(x)=L and limit an =L theorem 5 follwing six squences converage to be limit limit lnn\n =0 ,limit (1+x/n)n=ex .... Based on Theorems 1 to 5 in Section 10.1...
Please prove this, thanks! 2. Let {xn n21 be a sequence in R such that all n > 0. If ( lim supra) . (lim supー) = 1 Tn (here we already assume both factors are finite), prove that converges.
3. Let (p) be a sequence of orthogonal functions on [a, b] having the property that the zero function is the only con- tinuous real-valued function f satisfying fo, dA ofor all nE N. Prove that the system (p.) is complete. (Hint: First use the hypothesis to prove that if fE P((a, b) satisfies fo, dA -0 for all n E N. then f = 0 a.e. Next use complteness of to prove that Parseval's equality holds for every fE...