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}

Expert Solution

Colby Messinger answered 1 year ago

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...

Let {an} be a bounded sequence. In this question, you will prove that there exists a convergent subsequence. Define a crest of the sequence to be a term am that is greater than all subsequent terms. That is, am > an for all n > m (a) Suppose {an} has infinitely many crests. Prove that the crests form a convergent subsequence. (b) Suppose {an} has only finitely many crests. Let an1 be a term with no subsequent crests. Construct a...

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.

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.

1. Prove that if a set A is bounded, then A-bar is also bounded. 2. Prove...

1. Prove that if a set A is bounded, then A-bar is also bounded. 2. Prove that if A is a bounded set, then A-bar is compact.

Prove that if f is a bounded function on a bounded interval [a,b] and f is...

Prove that if f is a bounded function on a bounded interval [a,b] and f is continuous except at finitely many points in [a,b], then f is integrable on [a,b]. Hint: Use interval additivity, and an induction argument on the number of discontinuities.

5. Let n = 60, not a product of distinct prime numbers. Let Bn= the set...

5. Let n = 60, not a product of distinct prime numbers. Let Bn= the set of all positive divisors of n. Define addition and multiplication to be lcm and gcd as well. Now show that Bn cannot consist of a Boolean algebra under those two operators. Hint: Find the 0 and 1 elements first. Now find an element of Bn whose complement cannot be found to satisfy both equalities, no matter how we define the complement operator.

. Let f(x) = 3x^2 + 5x. Using the limit definition of derivative prove that f...

. Let f(x) = 3x^2 + 5x. Using the limit definition of derivative prove that f '(x) = 6x + 5 Then, Find the tangent line of f(x) at x = 3 Finally, Find the average rate of change between x = −1 and x = 2

Prove that a subspace of R is compact if and only if it is closed and bounded.

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