Let {an}n≥1 be a convergent sequence of real numbers. 1) Show that if for all but finitely many an we have an ≥ a, then limn→∞ an ≥ a. 2) Show that if for all but finitely many an we have an ≤ b, then limn→∞ an ≤ b. 3) Conclude that if all but finitely many an belong to the interval [a, b], then limn→∞ an ∈ [a, b].

Q: Exercise 10. Show that if AC B, then ACB.

A:

Q: A sequence (xn)∞ {n=1} is said to oscillate if lim inf xn < lim sup xn.  Prove or disprove the…

A: We can solve this question using definition of converges diverges and oscillating of sequences

Q: Show that x, converges to x.

A: Given that X be a metric space . Let xn be the sequence in a metric space X. Suppose that xn has a…

Q: 2. Let ao E R be an initial point. For each n > 0, we define Зал if an 0.5. (a) Prove that if ao…

A: It is given that an+1=3an            if  an≤0.5-3an+3   if an&gt;0.5.

Q: 1. Prove that the series 7n (a) converges uniformly on every bounded interval (b) does not converge…

A: As per norms . three questions ,1a, 1b and 2 are answered. To analyze various aspects of convergence…

Q: For a convergent real sequence sn and a real number a, show that if sn ≥ a for all but finitely many…

A:

Q: 1 1 Зп — 1 3n + 2 n=1

A:

Q: (22) Wedenote by the space of all bounded sequences (a,)-1 For example, (1, -2,1, –2, 1,–2, ...) E .…

A:

Q: Theorem 2.13: Let {an} be a sequence. Then the number L is a finite subsequential limit of {an} if…

A: Let an be a sequence.  (i) Given that there a infinitely many terms of an that equal L. Define a set…

Q: Proposition 4.6.10. Let f : R → R be defined by f(x) = +b where a andbare constants. Then f is = ax…

A: Given: fx=ax+b Let c be any arbitrary point in ℝand xn:n∈ℕ be a sequence in ℝ such that xn→c So,…

Q: Given (an) be a sequence of real numbers. Prove that if lim sup |an| = a for some a ∈ R, then (an)…

A: Limit superior: The greatest limit point of a bounded sequence is called limit superior. It is…

Q: 3. Prove that a sequence {a,} converges to the real number a if and only if lim sup a lim inf an =…

A: First suppose sequence  an is converges to a Then we have to prove that limsupan=liminfan=a Fix an…

Q: Assume A is a bounded, non-empty subset of R. Let r = sup(A). Prove there is a sequence from A…

A: Given r=sup(A). Let r∈A.  Consider the sequence {xn} such that xn=r ∀n∈ℕ.Then {xn} is a sequence…

Q: Let 1 ≤ p < ∞ and let {fn} be a sequence of functions in Lp[a, b]. Suppose there is a function f ∈…

A: We are given a sequence of functions {fn} in Lp[a,b].

Q: Let a, be a sequence of real numbers such that an+1 2 an, Vn 21 and a, + a as n + o. Use this…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: (10) Let H be a Hilbert space and let {an} be a sequence in H. We say that {an} converges weakly to…

A:

Q: 2. Prove that the sequence {n} defined by xn := (-1)n 3n+2 is divergent. Use an argument similar to…

A:

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: (ii) If (sn) is an unbounded decreasing sequence, then lim så -∞

A: Given that {sn} is an unbounded decreasing sequence. To show :- limn→∞sn=-∞

Q: Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all but finitely many n, then lim…

A: Let (sn) be a convergent sequence, and assume that sn ≥ a for all but finitely many n.

Q: Let {fn} be a sequence of funtions in L[a, b]. Suppose that there exists a function g ∈ L[a, b] with…

A: Given, limn→∞fnx=fx in a,b and hx is a bounded measurable function.

Q: Let an be recursively defined by ao = 2, an+1 = √5an for n ≥ 0. Either show that an converges and…

A:

Q: Let f: X →Y be an injective function, and A, B C X. Prove that Q1) f(A)n f(B) C f(AN B).

A:

Q: Let f be a function from the complex numbers to the reals. Show that if for every convergent…

A: Definition: Let an be a convergent sequence that converges to the limit l, if given ε&gt;0, there…

Q: If H be a Hilbert space and C be a subset of H. Give an example of a function, T:C- C, which…

A: Given that H is a Hilbert space and C is a subset of H. To find  a function that satisfies the…

Q: Let {an}1 be a sequence. (a) Suppose that {an}1 is convergent. Prove that limn→∞ an+1 exists and lim…

A:

Q: Let an be a bounded sequence of real numbers. Let s = sup{an: n ≥ 1} {ann € N}. Show that there is a…

A:

Q: (3.1) Let X be a topological Hausdorff space. Let (rn)n be a sequence in X which converges to c. Let…

A: Recall that, a set in a topological space is said to be compact if every open cover of a set has a…

Q: 1* Let X be the space of sequences of real numbers with only finitely many nonzero terms, with the…

A:

Q: a) Give an example of a monotone divergent sequence.

A:

Q: 15. Let {fn} be a sequence on C([a, b]). Assume that fn (x) 2 fn+1(x) for all x E [a, b]. Show that…

A:

Q: Suppose (an) is a sequence of positive real numbers such that an+1 Prove that liman (an) even…

A:

Q: Let ℝ∞ = {(x1, x2, ...)|xi is real for all i} be the set of all real sequences. Define addition as…

A: Use the definition of vector space:   A vector space is a nonempty set V with two operations called…

Q: is a bounded sequence in H and since A is a compact operator, there is a subsequence (Yn₁) such that…

A: We need to explain this portion of the proof:

Q: 14. Let {f} be a sequence on C(X) where X CR is compact. Assume that fn (x) ≥ fn+1(x) for all x E X.…

A: Given:A sequence of functions  in the space of complex-valued continuous functions , where  is a…

Q: HW12 Challenge: Prove this theorem by contradiction. You have three given lines and you can use the…

A:

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.