Suppose that a sequence {Zn} satisfies Izn+1-Znl < 2-n for all n e N. Prove that...
1. Prove that if {xn} is a sequence that satisfies 2n² + 3 Xnl73 +5n2 + 3 + 1 for all n e N, then {xn} is Cauchy. . Use the definition of limit for a sequence to show that 2. Suppose that {Xn} converges to 1 as n xn +1-e, as nº n
- a) Let Xn be a sequence such that Xn+1 – xn| son for all n E N. Show that the sequence is Cauchy (and hence convergent). b) Is the result in part a) true if we assume that In+1 – 2n| <
Let X1, X2,...be a sequence of random variables. Suppose that Xn?a in probability for some a ? R. Show that (Xn) is Cauchy convergent in probability, that is, show that for all > 0 we have P(|Xn?Xm|> )?0 as n,m??.Is the converse true? (Prove if “yes”, find a counterexample if “no”)
4 Consider the sequence () defined by, (a) Using 2, find r2 and r3 and express the results as true rational numbers. (b) Use induction to show that if xi є Q, then xnE Q for all n є N. (c) Prove, using induction, that if 2 x1 3, then 2 xn 3 for all n є N by showing i) 2 < rn < 3 implies that n+13 ii 2 S n 5/2 implies that 2 n+ i) 5/2...
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....
(15 points) Suppose that a sequence {{n}.00, of real numbers satisfies 52n+1 = 3xn + 2 for all n E N. Show that {{n}", converges. What is limnto Xn? Explain why? the following four nrohlems
Suppose that k e N and that f R"-R is homogeneous of order k: that is, that 'f(px)- kf(x) for all x є Rn and all є R. If f is differentiable on R", prove that af Экп af axi (Xi , . . . , xn) ER". for all x Suppose that k e N and that f R"-R is homogeneous of order k: that is, that 'f(px)- kf(x) for all x є Rn and all є R. If...
4. RWI 4.5.13). Suppose that the sequence (xn) satisfies n 1,2,.. X2ax.1 + bx and that 0< r<R satisfy - az- b (z-rz- R) (a) Show that x O(R" ). O(2") is false. (b) Give an example with r R 2 for which x, (c) What asymptotic (big-oh) estimate for (x) can you give in general if r R> 0?
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...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.) Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...