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Exercise 2.3.9. (a) Let (an) be a bounded (not necessarily convergent) sequence, and assume lim bn...
Let (an)nen be a bounded sequence in R. For all n e N define bn =...
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: Let (w).>o be a sequence such that bn is convergent. Let (an)nzo to be...
Problem 3: Let (w).>o be a sequence such that bn is convergent. Let (an)nzo to be a sequence such that lant - and by for all ne N. Prove that (an)nzo is a convergent sequence. (Hint: We did something similar in class before.)
***You must follow the comments*** Topic: Mathematical Real Analysis - Let (xn) be a bounded sequence...
***You must
follow the comments***
Topic: Mathematical
Real Analysis
- Let (xn) be a
bounded sequence ((xn) is not necessarily convergent), and assume
that yn → 0. Show that lim n→∞ (xnyn) = 0.
Question1.
All the
solution state that there exists M >0 and xn<=M . My question
is that why M always be bigger than 0 and Why it is bounded above ?
why it is not m<=xn bounded below????
Question.
2.
if the
sequence is convergent, then...
13 14 Exercise 13: Let (xn) be a bounded sequence a S be the set of...
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). Form...
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...
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,...
Problem 4 Let lim an = A, lim bn = B. Show that limah = AB...
Problem 4 Let lim an = A, lim bn = B. Show that limah = AB Proceed by following these steps. (a) Verify that | ab. - AB| = |a,(6,- B)+a,B - AB S 14,1-16,- B+B |a, -Al<e bounded small constant small Why is it possible to assume that "bounded times small + constant times small" is less than epsilon? (b) Let M be such that Jan <M Wn. Let Ni be such that 12x - Al <10+1 Yn >...
Let (an) be a sequence such that lim an = 0. Define the sequence (AR) Exercise...
Let (an) be a sequence such that lim an = 0. Define the sequence (AR) Exercise 21: by A =ļa, and An = zou-a + ax=a + zam for k21. Prove that an converges to some S if and only if Ax converges to S. N=0 k=0 Exercise 22: (Cauchy condensation test) Let (an) be a sequence such that 0 < antı san a) Show n=0 n=1 Hint: Recall the proof of convergence of for p > 1. Ren for...
Q6 6 Points Let an and bn be 2 convergent sequences. Let A = limno an...
Q6 6 Points Let an and bn be 2 convergent sequences. Let A = limno an and B = limno br. Prove that limno anbn AB = You may use the following inequality anbn - AB B) + Blan - A), exercise 4, triangular inequalities, among other results proved in class or above. : an (bn
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 define...
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...
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: Let (w).>o be a sequence such that bn is convergent. Let (an)nzo to be a sequence such that lant - and by for all ne N. Prove that (an)nzo is a convergent sequence. (Hint: We did something similar in class before.)
***You must
follow the comments***
Topic: Mathematical
Real Analysis
- Let (xn) be a
bounded sequence ((xn) is not necessarily convergent), and assume
that yn → 0. Show that lim n→∞ (xnyn) = 0.
Question1.
All the
solution state that there exists M >0 and xn<=M . My question
is that why M always be bigger than 0 and Why it is bounded above ?
why it is not m<=xn bounded below????
Question.
2.
if the
sequence is convergent, then...
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...
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,...
Problem 4 Let lim an = A, lim bn = B. Show that limah = AB Proceed by following these steps. (a) Verify that | ab. - AB| = |a,(6,- B)+a,B - AB S 14,1-16,- B+B |a, -Al<e bounded small constant small Why is it possible to assume that "bounded times small + constant times small" is less than epsilon? (b) Let M be such that Jan <M Wn. Let Ni be such that 12x - Al <10+1 Yn >...
Let (an) be a sequence such that lim an = 0. Define the sequence (AR) Exercise 21: by A =ļa, and An = zou-a + ax=a + zam for k21. Prove that an converges to some S if and only if Ax converges to S. N=0 k=0 Exercise 22: (Cauchy condensation test) Let (an) be a sequence such that 0 < antı san a) Show n=0 n=1 Hint: Recall the proof of convergence of for p > 1. Ren for...
Q6 6 Points Let an and bn be 2 convergent sequences. Let A = limno an and B = limno br. Prove that limno anbn AB = You may use the following inequality anbn - AB B) + Blan - A), exercise 4, triangular inequalities, among other results proved in class or above. : an (bn
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...
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