Show that the set of limit-points of a sequence is a closed set.

Q: Give an example of a bounded sequence {a„}_1 having exactly 2 limitpoints.

A: we have to find a sequence such that it has two limit points.

Q: Determine the limit of the sequence or show that the sequence diverges by using the appropriate…

A: Given:   To find: The limit of C n  if it exist or to show the sequence diverges by appropriate…

Q: limit of this sequence

A: We have to find out the limit of the given sequence.

Q: O Show that every Cauchy sequence in RP is bounded.

A: To show that every cauchy sequence is bounded.

Q: lim nsin(2/n) 2→+∞

A: Given query is to find the limit of the sequence.

Q: Every bounded sequences is convergent.

A:

Q: Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?

A:

Q: how that every Cauchy sequence {pn} is bounded.

A:

Q: Give an example of a bounded sequence without a limit.

A: If any sequence has a limit, then that sequence is said to be convergent. So here the objective is…

Q: (a) Prove that the sum of the degree sequence of a simple graph is an even integer.

A: Given below the proof

Q: Prove that y is a negative Cauchy sequence if and only if -y is said to be a positive Cauchy…

A: We need to prove that y is a negative Cauchy sequence if and only if -y is a positive Cauchy…

Q: Find the limit of the fallowing sequence if it exists 92 = 12-2, an =/2-/2+12+2+..+2 ' n eN

A: Explained below

Q: assume that each sequence converges and findits limit. a1 = 5, an+1 = √5an

A:

Q: Let {bn} be a sequence whose limit is 3. Show that if B is an upper bound of this sequence, then 3 <…

A: We will use the definition of limit.

Q: Find an example of two unbounded sequences , such that their product gives a bounded sequence.

A: Consider the two unbounded sequences

Q: Exercise 4. Use the definition of limit to prove that the sequence {n – - is divergent. -

A: To prove: the sequence n-1nn=1∞ is divergent. limn→∞n-1n=limn→∞n-limn→∞1n…

Q: What is a monotonic sequence? Under what circumstances does such a sequence have a limit? Give…

A: The monotonic sequence theorem:

Q: c) Prove that a sequence in R can have at most one limit that is “uniqueness of limits"

A: consider  Xn be a sequence in ℝ

Q: 3. Use the definition of limits of sequence to give a formal proof of lim (√n+1-√n) = 0.

A:

Q: Give an example of a bounded sequence that has a limit and an example of a bounded sequence that…

A: To give an example of a bounded sequence that has a limit and an example of a bounded sequence that…

Q: Suppose you know that  {an} is a decreasing sequence andall its terms lie between the numbers 5 and…

A: Given: {an}is a decreasing sequence &amp; all the terms of sequence lies between 5 &amp; 8. To…

Q: Prove that the following sequence is convergent using the limit definition

A: Solve the following

Q: Give an example of a bounded sequence that has a limit.

A: Give an example of a bounded sequence that has a limit.

Q: consider the sequence  1, 1/2 ,1, 1/3 ,1 ,1/4, 1,.........  determine the limit superior and limit…

A: Given sequence 1, 1/2 , 1 , 1/3.. The limit superior is 1. As every number of sequence &lt; 1+k for…

Q: Every convergent sequence of real numbers is not bounded ylgn İhi

A: Every convergent sequence of real numbers is not bounded.

Q: If {b,}-1 is a bounded sequence and lim,→∞ An = lim,-+∞ anbn? Justify. 0, what can you say about

A: Given that  bn is bounded sequence, so, by definition of bounded sequence  ∃B&gt;0 such that bn&lt;B…

Q: Let {an} be a monotone sequence. Show that (a,} is a Cauchy sequence if and only if it is bounded.

A:

Q: Evaluate this limit of sequence

A: The limit of a sequence can be found as  limn→∞an that is how the sequence behave when n tends to…

Q: Show that the converse of Theorem Limit of a Sequence is not true.

A: The theorem of Limit of a Sequence states that, " The function f be a function of a real number…

Q: Using the definition of limit of a sequence, show that n+1 lim 1.

A:

Q: (-1)" 50. Сп

A: To show that cn=(-1)n/√n diverges

Q: Prove that a bounded, strictly increasing sequence converges.

A: To prove that a bounded, stritly increasing sequence converges

Q: b) Set up In rdr as a limit of sums.

A: In this question, concept of Riemann Sum is applied. Riemann Sum To estimate ∫abfxdx , Riemann sums…

Q: Consider the sequence  1,1/2,1,1/3,1,1/4,1, ... Determine the limit superior and limit inferior of…

A: Given sequence is  1,12,1,13,1,14,1,....... By observation , it can be seen clearly that 1 repeats…

Q: 2. Prove that every bounded sequence in a metric space has a convergent subsequence

A: To prove that every bounded sequence in a metric space has a convergent subsequence. It is enough to…

Q: Determine the limit of the sequence defined below. an 13 n+5 If the limit does not exist, enter Ø.…

A: limn→∞-113nnn+5 =limn→∞(-1)n113nnn+5 let n is even (-1)n=(-1)2m=1 L=limn→∞113nnn+5=0*1=0 let n is…

Q: Evaluate the limit of sequence ak  = k-1/k-2

A: We have to find limit

Q: Consider the sequence  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, ... Determine the limit…

A: Here sequence is given derive from.this

Q: Please give an cxample of a sequence (a,) such that: (a) (an) is not bounded above. (b) (an) is not…

A:

Q: Please give an cxample of a sequence (a,) such that: (a) (an) is not bounded above. (b) (an) is not…

A:

Q: In(n) In(2n).

A:

Q: Show that the limit of a convergent sequence in a metric space is unique

A:

Q: What is the connection between the concept of the limit of a sequence and the limit of the function?

A: Solution

Q: Determine if the sequence  {8/1 , 8/3 , 4/2 , 4/4 , 8/5 , 8/7 , 4/6 , 4/8 , . . .}  converges and,…

A: Given, Sequence S=81,83,42,44,85,87,46,48,...

Q: of all sequences that are "eventually zero," that is, al

A:

Q: What theorems are available for calculating limits of sequences? Give examples.

A: There are six theorems are available for calculating limits of sequence.

Q: Give a sequence of positive numbers which is NOT bounded and has exactly one limit point.

A: A real no.p is said to be a limit point if every neighborhood of p contains an infinite no. Of…

Q: Prove that every bounded below, decreasing sequence converges

A: Let {an} be a monotonically decreasing sequence which is bounded below.  Since the sequence is…