Solutions to Tutorial 3 (Week 4) Material covered Outcomes PDF

Sequences

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.

Lecture 2 : Convergence of a Sequence Monotone sequences

Solutions to Tutorial 3 (Week 4) Material covered Outcomes

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.

MAT 319 Spring 2012 Solutions to HW 4 1. Prove the following: (a

(a) If the sequence (xn) is bounded below and (yn) diverges to +?

5 Sequences

(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

Solutions Math 201A: Fall 2016 Problem 1. Let (X d) be a metric

(b) Prove that if xn ? x and yn ? y as n ? ? then d(xn

2.6: The Cauchy Criterion

subsequence of a convergent sequence is convergent this is impos- a) If (xn) and (yn) are Cauchy sequences

Week 2 Solutions Page 1 Exercise (2.1.3). Is the sequence

Let {xn} and {yn} be bounded sequences. a) Show that {xn + yn} is bounded. b) Show that. (lim inf n

Real Analysis Math 127A-B Winter 2019 Midterm 1: Solutions 1

(b) Every bounded monotonic sequence is a Cauchy sequence. (c) If the sequences (xn) and (yn) diverge then the sequence (xnyn) diverges.

Metric Spaces 2000 Lecture 13

n=1 be Cauchy sequences in a metric space (X d). Then lim n?? d(xn

5 Sequences - Pennsylvania State University

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)

Contents Introduction to Sequences - University of Chicago

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

MATH 409 Summer 2019 Practice Problem Set 2 - Texas A&M

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

Sequences I - The University of Warwick

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

Sequences - UC Davis

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

Searches related to sequences xn and yn filetype:pdf

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.

[PDF] Solutions to Tutorial 3 (Week 4) Material covered Outcomes

What is an example of a sequence?

What if (xn)1 =1converges to some '2R?

Is there a convergent subsequence of x n?

How to prove that (xn)1 =1 is convergent?