Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all...

Let (s_n) be a sequence that converges.

(a) Show that if s_n ≥ a for all but finitely many n, then lim s_n ≥ a.

(b) Show that if s_n ≤ b for all but finitely many n, then lim s_n ≤ b.

(c) Conclude that if all but finitely many sn belong to [a,b], then lim s_n belongs to [a, b].

Expert Solution

Here, first we prove the first part , further using this we prove the part (b), (c).

of a and b are disjoint.

Now part b:

Now part d:

Thus we are done!

Colby Messinger answered 1 year ago

Let {λn} be a sequence of scalars that converges to zero, limn→∞ λn = 0. Show...

Let {λn} be a sequence of scalars that converges to zero, limn→∞ λn = 0. Show that the operator A : ℓ2 → ℓ2 , A(x1, x2, ..., xn, ...) = (λ1x1, λ2x2, ..., λnxn, ...) is compact. What is the spectrum of this operator?

Show that a monotone sequence converges if and only if it is bounded.

Let S = (s1, s2, . . . , sn) be a given sequence of integer...

Let S = (s1, s2, . . . , sn) be a given sequence of integer numbers. The numbers can be positive or negative. We define a slice of S as a sub- sequence (si,si+1,...,sj) where 1 ≤ i < j ≤ n. The weight of a slice is defined as the sum of its elements. Provide efficient algorithms to answer each of the following questions: a)Is there any slice with zero weight ? b)Find the maximum weight slice in...

Let sn be a Cauchy sequence such that ∀n > 1, n ∈ N, ∃m >...

Let sn be a Cauchy sequence such that ∀n > 1, n ∈ N, ∃m > 1, m ∈ N such that |sn − m| = 1/3 (this says that every term of the sequence is an integer plus or minus 1/3 ). Show that the sequence sn is eventually constant, i.e. after a point all terms of the sequence are the same

prove every cauchy sequence converges

Prove that every sequence in a discrete metric space converges and is a Cauchy sequence. This...

Prove that every sequence in a discrete metric space converges and is a Cauchy sequence. This is all that was given to me... so I am unsure how I am supposed to prove it....

Abstract Algebra Let n ≥ 2. Show that Sn is generated by each of the following...

Abstract Algebra Let n ≥ 2. Show that Sn is generated by each of the following sets. (a) S1 = {(1, 2), (1, 2, 3), (1, 2, 3, 4), ..., (1, 2, 3,..., n)} (b) S2 = {(1, 2, 3, ..., n-1), (1, 2, 3, ..., n)}

E.C. 2. (10 pts.) Suppose that (sn) is a sequence of real numbers such that sn...

E.C. 2. (10 pts.) Suppose that (sn) is a sequence of real numbers such that sn ≥ 0 for all n ∈ N. (a) Show that the set of subsequential limits of S satisfies S ⊆ [0,∞) ∪ {+∞}. (b) Is it possible for S = [0,∞) ? (Hint: apply Theorem 11.9.) Legible handwriting is a must

Let τ ∈ Sn be the cycle (1, 2, . . . , k) ∈ Sn...

Let τ ∈ Sn be the cycle (1, 2, . . . , k) ∈ Sn where k ≤ n. (a) For σ ∈ Sn, prove that στσ-1 = (σ(1), σ(2), . . . , σ(k)). (b) Let ρ be any cycle of length k in Sn. Prove that there exists an element σ ∈ Sn so that στσ-1 = ρ.

Let sn = 21/n+ n sin(nπ/2), n ∈ N. (a) List all subsequential limits of (sn)....

Let sn = 21/n+ n sin(nπ/2), n ∈ N. (a) List all subsequential limits of (sn). (b) Give a formula for nk such that (snk) is an unbounded increasing subsequence of (sn). (c) Give a formula for nk such that (snk) is a convergent subsequence of (sn).

Question