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

Expert Solution

Colby Messinger answered 2 years ago

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

Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all but finitely many n, then lim sn ≥ a. (b) Show that if sn ≤ b for all but finitely many n, then lim sn ≤ b. (c) Conclude that if all but finitely many sn belong to [a,b], then lim sn belongs to [a, b].

(a) a sequence {an} that is not monotone (nor eventually monotone) but diverges to ∞ (b)...

(a) a sequence {an} that is not monotone (nor eventually monotone) but diverges to ∞ (b) a divergent sequence {an} such that {an/33} converges (c) two divergent sequences {an} and {bn} such that {an + bn} converges to 17 (d) two convergent sequences {an} and {bn} such that {an/bn} diverges (e) a sequence with no convergent subsequence (f) a Cauchy sequence with an unbounded subsequence

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 every sequence contains a monotone subsequence and explain how this furnished a new proof...

Show that every sequence contains a monotone subsequence and explain how this furnished a new proof of the Bolzano-Weierstrass Theorem.

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....

Show that if a preference relation is locally nonsatiated then it is not monotone

A sequence (xn) converges quadratically to x if there is some Q ∈ R such that...

A sequence (xn) converges quadratically to x if there is some Q ∈ R such that |xn − x| ≤ Q/n^2 for all n ∈ N. Prove directly that if (xn) converges quadratically, then it is also Cauchy.

Let sequence an be a bounded sequence and let E be the set of subsequential limits...

Let sequence an be a bounded sequence and let E be the set of subsequential limits of an. prove that E is bounded and contains both sup E and inf E

If the sequence is increasing then it a) converges to its supremum b) diverges c) may...

If the sequence is increasing then it a) converges to its supremum b) diverges c) may converge to its supremum d) is bounded if S= { 1/n - 1/m: n,m belongs to N } where N is the set of natural numbers then infimum and supremum of S respectively are a) -1and 1 b) 0,1 c)0,0 d)can not be determined Please explain

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