If xn = −1, yn = 1, and zn = (−1)n+1, then xn ≤ zn ≤ yn for all n ∈ N, the sequence (xn) converges to −1 and (yn) converges 1, but (zn) does not converge As
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(ii) If (xn) and (yn) are convergent sequences, then the sequence (xnyn) converges and lim(xnyn) = lim xn limyn (iii) If (xn) converges to x and x = 0, then almost all
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For example, ((−1)n) is a bounded sequence but it does not converge Limit rules for convergent sequences: Theorem 3 Let xn → x and yn → y Then (a) xn +
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The following three results enable us to evaluate the limits of many sequences Limit Theorems Theorem 2 1: Suppose xn → x and yn → y Then 1 xn + yn → x
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Week 2 Solutions Page 2 Exercise (2 3 7) Let {xn} and {yn} be bounded sequences a) Show that {xn + yn} is bounded b) Show that (lim inf n→∞ xn) + ( lim inf
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29 mai 2019 · yn 3 the sequence (xn · yn)∞ n=1 is convergent and that lim
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Proof: Since (xn) diverges to infinity, we can always find a tail of the sequence whose terms are as large as we want In particular, ∃N such that xn > 1000
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(For this exercise the Algebraic Limit Theorem is off-limits, so to speak ) Exercise 2 3 3 (Squeeze Theorem) Show that if xn ≤ yn ≤ zn for all n ∈ N, and if lim
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Theorem (3 2 5) If (xn) and (yn) are convergent sequences and if xn ≤ yn ∀ n ∈ N, then lim(xn)
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Limits of convergent sequences preserve (non-strict) in- equalities. Theorem 3.22. If (xn) and (yn) are convergent sequences and xn ? yn for all n ? N then.
Let (xn) and (yn) be bounded sequences in R with xnyn ? 0 for all n ? N. (a) Show that lim sup n??. (xnyn) ? (lim sup.
(a) If the sequence (xn) is bounded below and (yn) diverges to +?
(ii) If (xn) and (yn) are convergent sequences then the sequence (xnyn) converges and lim(xnyn) = lim xn limyn. (iii) If (xn) converges to x and x = 0
(b) Prove that if xn ? x and yn ? y as n ? ? then d(xn
subsequence of a convergent sequence is convergent this is impos- a) If (xn) and (yn) are Cauchy sequences
Let {xn} and {yn} be bounded sequences. a) Show that {xn + yn} is bounded. b) Show that. (lim inf n
(b) Every bounded monotonic sequence is a Cauchy sequence. (c) If the sequences (xn) and (yn) diverge then the sequence (xnyn) diverges.
n=1 be Cauchy sequences in a metric space (X d). Then lim n?? d(xn
If (xn) is a null sequence and (yn) is a bounded sequence then thesequence (xnyn) is a null sequence If (xn) and (yn) are convergent sequences then the sequence (xnyn)converges and lim(xnyn) = limxnlimyn If (xn) converges toxandx6= 0 then almost all terms of (xn) arenonzero and the sequence (1/xn) converges to 1/x Proof of (iI)
convergence and divergence bounded sequences continuity and subsequences Relevant theorems such as the Bolzano-Weierstrass theorem will be given and we will apply each concept to a variety of exercises Contents 1 Introduction to Sequences 1 2 Limit of a Sequence 2 3 Divergence and Bounded Sequences 4 4 Continuity 5 5
1 be sequences of real numbers Assume that (x n)1 =1 is bounded and that lim n!1y n= 0 Let (z n) 1 n=1 = (x ny n) 1 =1 Show that lim n!1z n= 0 Hint Exploit the de nitions of convergence boundedness and the properties of the absolute value Possible solution Assume that there exists M 0 such that for all n2N jx nj M and that lim n!1y
Figure 2 4: Sequences bounded above below and both Bounds for Monotonic Sequences Each increasing sequence ( a n) is bounded below by a1 Each decreasing sequence ( a n) is bounded above by a1 Exercise 3 Decide whether each of the sequences de?ned below is bounded above bounded below bounded If it is none of these things then explain why
A sequence (xn) of real numbers is a functionf: N!R wherexn=f(n) We can consider sequences of many di erent types of objects (for examplesequences of functions) but for now we only consider sequences of real numbersand we will refer to them as sequences for short A useful way to visualize asequence (xn) is to plot the graph ofxn 2Rversusn2N
Di?erent sequences of convergent in probability sequences may be combined in much the same way as their real-number counterparts: Theorem 7 4 If X n ?P X and Y n ?P Y and f is continuous then f(X nY n) ?P f(XY) If X = a and Y = b are constant random variables then f only needs to be continuous at (ab)
What is an example of a sequence?
The frst example of a sequence is x n=1 n In this case, our function fis defned as f(n) =1 n As a listed sequence of numbers, this would look like the following: (1.3) 1; 1 2 ; 1 3 ; 1 4 ; 1 5 ; 1 6 ; 1 7 ::: Another example of a sequence is x
What if (xn)1 =1converges to some '2R?
(ii)If (xn )1 =1converges to some ‘2R, then ‘2Z. Hint. For i) use the defnition of Cauchy sequence for a well chosen "and derive a contradiction if the sequence is not eventually constant.
Is there a convergent subsequence of x n?
By the Bolzano-Weierstrass Theorem, we can say that there indeed exists a convergent subsequence of x n. Just by looking at this sequence, we can see four convergent subsequences: (1,1,1...), (2,2,2...), (3,3,3...), and (4,4,4...). These subsequences con- verge to 1, 2, 3, and 4 respectively.
How to prove that (xn)1 =1 is convergent?
Let (xn )1 =1be a sequence of real numbers. Show without using the Monotone Convergence Theorem that if (xn )1 =1is decreasing and bounded below then (xn )1 =1is convergent. Hint. You could mimic the proof of the increasing version and use the ap- proximation property of infma to show that (xn )1 =1converges to inffx n: n2 Ng. Possible solution.
Sequences