(n2−1 n2 ) is Cauchy using directly the definition of Cauchy sequences Proof Given ϵ > 0 Let {xn} be a sequence such that there exists a 0
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Let X = (xn) be a sequence of positive real numbers such that lim (xn+1 xn ) is Cauchy 4 Let (fn) ∈ C[0,1] be such that there exists M > 0 such that fn ∞ ≤
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A real sequence (xn) is called contractive if there exists a constant 0 0 and let N be such that x2 − x1 Cn−1 1 1−C
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(c) If (xn) satisfies the Cauchy criterion, then there exists an α ∈ R such that 0
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every monotone increasing bounded sequence bounded above converges that for every ε > 0 there exists some n > 1/ε such that xn < ε 1 (c) Using the Archimedean property, argue that yk cannot be bounded above by M, hence
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If xn → 0, then there exists n0 ∈ N such that xn < 1 2 If possible, let there exist a non-convergent sequence Hence there exists c ∈ (0,∞) such that f (c) = 0
Practice Solution
(xn)∞ n=0 is a function f : N0 → R where xn = f(n) and N0 = {0, 1, 2, 3, }, and eventually; and (xn) does not converge to x ∈ R if there exists ϵ0 > 0 such that We let x = lim n→∞ xn, y = lim n→∞ yn The first statement is immediate if c = 0
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(c) A divergent monotone sequence with a Cauchy subsequence Let (an) and (bn) be Cauchy sequences Decide whether each k=1 ak is Cauchy if and only if for all ϵ > 0 there exists N ∈ N such that whenever n>m ≥ N (b) A convergent series ∑xn and a bounded sequence (yn) such that ∑xnyn diverges (c) Two
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29 mai 2019 · Show that a sequence (xn)∞ n=1 is convergent to l ∈ R, if and only if for every ε ∈ (0, 2) there exists N ∈ N such that for all n ⩾ N, xn − l < ε
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(n2?1 n2. ) is Cauchy using directly the definition of. Cauchy sequences. Let {xn} be a sequence such that there exists a 0 <C< 1 such that.
Show that X is not a bounded sequence and hence is not convergent. Solution. Since lim. (xn+1 xn. ) = L given ? > 0
Suppose xn is a bounded sequence in R. ?M such that but there are no such pt. ... Let V = C([01])=all continuous functions on the interval.
Suppose that (xn) is any sequence in A with xn = c that converges to c and let ? > 0 be given. From. Definition 6.1
Proposition 3.19. A convergent sequence is bounded. Proof. Let (xn) be a convergent sequence with limit x. There exists N ? N such that.
(b) Let E ? R be a subset such that there exists a sequence {xn} in E with Solution: For any x = 0 there exists an N such that
converges to 0 then the sequence (xn n) must converge to 0. Solution: The given statement is TRUE. If xn ? 0
continuous at c if for every ? > 0 there exists a ? > 0 such that In particular f is discontinuous at c ? A if there is sequence (xn) in the domain.
Let us now state the formal definition of convergence. Definition : We say that a sequence (xn) converges if there exists x0 ? IR such that for every.
(c) If {xn} is a sequence of real (or complex) numbers that converges to Now suppose {xn} converges to x i e for all ? > 0 there exists N ? N such
Proposition 3 19 A convergent sequence is bounded Proof Let (xn) be a convergent sequence with limit x There exists N ? N such
xn = s Proof Let ? > 0 be given Since limn?? an = s there exists a positive integer N1 such that
Suppose xn is a bounded sequence in R ?M such that but there are no such pt Let V = C([01])=all continuous functions on the interval
converges to 0 then the sequence (xn n) must converge to 0 Solution: The given statement is TRUE If xn ? 0 then there exists n0 ? N such that xn < 1
Let us now state the formal definition of convergence Definition : We say that a sequence (xn) converges if there exists x0 ? IR such that for every
(b) Let E ? R be a subset such that there exists a sequence {xn} in E with the property that xn ? x0 /? E Show that there is an unbounded continuous
Let X = (xn) be a sequence of positive real numbers such that lim (xn+1 xn ) Let (fn) ? C[01] be such that there exists M > 0 such that fn ? ?
Let A be a bounded subset of R Show that there exists a sequence (an) of elements of A such that lim(an) = sup(A)
Let {an} be a bounded sequence such that every convergent subsequence of {an} has a limit L Prove that limn?? an = L Solution Method 1: Note that La = {L}
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Week 3 Solutions Page 1 Exercise (2.4.1). Prove that ) is Cauchy