3. For the sequence AS compute a real number p such that lim an 7 Indicate...
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...
18. If ai, az, as,... is a bounded sequence of real numbers, define lim sup an (also denoted lim an) to be --+ n+ l.u.b. {z ER: an > & for an infinite number of integers n} and define lim inf an (also denoted lim an) to be g.l.b. {ER: An <for an infinite number of integers n}. Prove that lim inf an Slim sup an, with the equality holding if and only if the sequence converges. 19. Let ai,...
Please only do number 8.
7. Prove that if {zn} is a sequence with lim zn w and lim zn = w, then lim z 8. For the sequence of the previous exercise, prove that lim 1/zn = 1/w provided w 0.
a and an+1= 5an +3 for any natural (Total 5+10= 15 pts) 4. For a positive real number a, consider the sequence (an)1 defined by a1 number n. Answer each queestion. (a) Without using e-N argument, show that the sequence (an)1 converges. (5 pts) (b) Using definition of limits, i.e., using e-N argument, show that the sequence (an)1 is a convergent sequence. If it converges, determine also the limit (10 pts)
a and an+1= 5an +3 for any natural (Total...
3) Let (an)2- be a sequence of real numbers such that lim inf lanl 0. Prove that there exists a subsequence (mi)2-1 such that Σ . an, converges に1
real analysis
Find the limit of the sequence as n to or indicate that it does not converge en2 0 (0,0,1) O Does not converge 0 (0, 1, 7) 0 (0,0,0) Is it true that any unbounded sequence in RN cannot have a convergent subsequence? Please, read the possible answers carefully. 0 Yes, because any sequence in RN is a sequence of vectors, and convergence for vectors is not defined. o Yes, it is true: any unbounded sequence cannot have...
Claim: {(-1)"} does not converge to any real number a. Proof: Assume that the sequence converges; that is, assume that there is an a E R such that lim,--.(-1)" = a. Then, using E = 1, from the definition of convergence, we know that there exists an no such that |(-1)" - al < 1 for all n > no. Thus, for any odd integer nno, we have |(-1)" - al = 1-1-a[< 1, and for any even integer n>...
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...
Given that lim F(x) = 0 lim g(x) = 0 lim h(x) = 1 Jim P(x) = lim (x) = .. evaluate the limits below where possible. (If a limit is indeterminate, enter INDETERMINATE.) (a) lim [fix)] lim [F(x)] X (c) lim [h(x)]04) 8 [(x)] X lim P(x)] 20 X (1) lim "P(x) X Enhanced Feedback Please try again, keeping in mind that the indeterminate cases are 0.9, 03.00,60,1", and " - .. Need Help? Read It Talk tea Tutor...
3. Find the limits, using L'Hospital's Rule where appropriate. (a) lim 1 - cos (7) 10 (b) lim In (3x + 8) 1+ In (6x + 5) + 10