Let {an} and {bn} be bounded sequences. Prove that limit superior {an+bn} ≦ limit superior {an}...

Let {a_n} and {b_n} be bounded sequences. Prove that limit superior {a_n+b_n} ≦ limit superior {a_n} + limit superior{b_n}

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Colby Messinger answered 2 years ago

1: (an) and (bn) are bounded sequences: (a) prove that limsup(-an) = -liminf(an) (b) for any...

1: (an) and (bn) are bounded sequences: (a) prove that limsup(-an) = -liminf(an) (b) for any c>0, prove that limsup(can) = climsup(an) and liminf(can) = climinf(an) (c) prove that limsup(an+bn) ≤ (limsup(an)) + (limsup(bn)) and liminf(an+bn) ≥ (liminf(an)) + (liminf(bn)) (d) If an and bn are made of nonnegative terms, prove that limsup(anbn) ≤ (limsup(an)) x (limsup(bn)) and liminf(anbn) ≥ (liminnf(an)) x (liminf(bn)) (e) prove that limsup(an+1) = limsup(an) and liminf(an+1) = liminf(an)

Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn...

Let (xn), (yn) be bounded sequences. a) Prove that lim inf xn + lim inf yn ≤ lim inf(xn + yn) ≤ lim sup(xn + yn) ≤ lim sup xn + lim sup yn. Give example where all inequalities are strict. b)Let (zn) be the sequence defined recursively by z1 = z2 = 1, zn+2 = √ zn+1 + √ zn, n = 1, 2, . . . . Prove that (zn) is convergent and find its limit. Hint; argue...

For any two real sequences {an} and {bn}, prove that Rudin’s Ex. 5 We assume that...

For any two real sequences {an} and {bn}, prove that Rudin’s Ex. 5 We assume that the right hand side is defined, that is, not of the form ∞ − ∞ or −∞ + ∞. lim sup (an + bn) ≤ lim sup an + lim sup bn. Proof If lim sup an = ∞ or lim sup bn = ∞, there is nothing to prove

Let {an} be a bounded sequence. In this question, you will prove that there exists a...

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Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if...

Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if and only if every subsequence of (an) has a convergent subsequence.

Let A be a subset of all Real Numbers. Prove that A is closed and bounded...

Let A be a subset of all Real Numbers. Prove that A is closed and bounded (I.e. compact) if and only if every sequence of numbers from A has a subsequence that converges to a point in A. Given it is an if and only if I know we need to do a forward and backwards proof. For the backwards proof I was thinking of approaching it via contrapositive, but I am having a hard time writing the proof in...

1.Prove the following statements: . (a) If bn is recursively defined by bn =bn−1+3 for all...

1.Prove the following statements: . (a) If bn is recursively defined by bn =bn−1+3 for all integers n≥1 and b0 =2, then bn =3n+2 for all n≥0. .(b) If cn is recursively defined by cn =3cn−1+1 for all integers n≥1 and c0 =0, then cn =(3n −1)/2 for all n≥0. .(c) If dn is recursively defined by d0 = 1, d1 = 4 and dn = 4dn−1 −4dn−2 for all integers n ≥ 2, then dn =(n+1)2n for all n≥0.

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Let {an} and {bn} be bounded sequences. Prove that limit superior {an+bn} ≦ limit superior {an}...

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