6. Let {n} and {yn} be the following sequences that repeat in cycles of four: n = 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, ... and Yn 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2... Find (a) lim inf an + lim inf yn (b) lim inf(n + Yn) (c) lim supan + lim sup yn

Q: 3.) lim r/2

A: To find the limit of the function

Q: Find the limit.

A:

Q: Consider the sequence a. Write out the first five terms of the sequence. a1 = a2 = a3 = a4= a5 = 9.…

A: here we have to check whether the sequence is convergent or not.

Q: lim [(n+1)(n+ 2)...(n+n)]/" n

A:

Q: 2. Let {an} be a monotone sequence. Show that {a,} is a Cauchy sequence if and only if it is…

A:

Q: These are alternative definitions of the limit of sequences. Can you please help identify what is…

A: Given: limn→∞an=a if ∀ε>0∃N∈ℕ such that ∀n≥N  an-a3<ε To Find: Identify what is wrong with the…

Q: 6. Show that the following sequences of sets, {C& }, are non-increasing, (1.2.17), then find lim-»∞…

A: * SOLUTION :-  (6) Given sequence of sets , {CK}, non- increasing, (1.2.17),

Q: Let {ai} be a sequence in a metric space X,d. Prove that lim an = p if and only if lim d(an, p) =…

A:

Q: 3. Let (Sn) be a sequence of nonnegative real numbers, and suppose lim s, = 0. Prove that lim S, =…

A:

Q: Show that the shift map o : {0, 1}N → {0,1}N is continuous, where o(x) is e sequence obtained by…

A:

Q: Show that lim (1 + = e. lim

A: To prove:

Q: 6. Show that the equation 5 = x* has at least one real root. You may assume that the functions 5 and…

A: Roots of the equation

Q: an + an-1 An+1 Vn +1 (a) Find the limit of an as n → ∞ and give a proof. converges for a,n from part…

A: Given, an+1=an+an-1n+1 the sequence is all positive number. (a) we need to find the limit of an as…

Q: In one plane, draw the closed second quadrant Q, along with the first three terms of the sequences…

A: Given Q2¯ is the closed second quadrant.  Given sequences are  an=lnnbn=1+iπn4 First three terms of…

Q: Suppose {Pi} and {qi} are sequences of real numbers such that lim pn = p and lim qn = q. Give an e-N…

A: suppose pi and qi are sequences of real numbers such that   limn→∞pn=p  and limn→∞qn=qSo by…

Q: 3. Consider the sequence an = sin n = 1,2, 3, 4,. Find lim sup an, lim inf an. n-00 n00

A:

Q: 4. Find the following limits: + x -6 lim x-2 - lim r2- x2-4 x2 - 4 Note that the second limit above…

A:

Q: 1. Let a, = 1 and an+1 = 1+ 1+an (a) Find the first six terms of the sequence {an}. (b) Prove that…

A: As you asked multiple sub parts so I answered 1st three subpart.