1 Review and Practice Problems for Exam # 2

A sequence (xn) of real numbers converges if and only if it is a Cauchy sequence. Proof: Assume that (xn) converges to a real number XO' Let the.

1) Cauchy Convergence Criterion: A sequence (xn) is Cauchy if and

1) Cauchy Convergence Criterion: A sequence (xn) is Cauchy if and only if it is convergent. Proof. Suppose (xn) is a convergent sequence and lim(xn) = x.

Lecture summary

A sequence (an) is said to be a Cauchy sequence iff for any ? > 0 there of real numbers in terms of operations on the underlying Cauchy sequences ...

1.4 Cauchy Sequence in R

Every Cauchy sequence in R converges to an element in [ab]. Proof. Cauchy seq. A real vector space V is a set of elements called vectors

ANALYSIS I 9 The Cauchy Criterion

We say that (an) is a Cauchy sequence if for all ? > 0 Every real Cauchy sequence is convergent. ... n=0 be a sequence of real (complex) numbers.

Chapter 2. Sequences §1. Limits of Sequences Let A be a nonempty

Theorem 4.2. A sequence of real numbers is convergent if and only if it is a Cauchy sequence. Proof. Suppose that (xn)n=

Sequences

A sequence of real numbers converges if and only if it is a Cauchy sequence. Proof. First suppose that (xn) converges to a limit x ? R. Then for every ? > 0.

1 Review and Practice Problems for Exam # 2 - MATH 401/501

Know the ? N definition of Cauchy sequences and of convergent sequences in R to A sequence of real numbers converges if and only if the limsup and the ...

1 Review and Practice Problems for Exam # 2 - MATH 401/501

Know the ? N definition of Cauchy sequences and of convergent sequences in R to A sequence of real numbers converges if and only if the limsup and the ...

1 Sequence and Series of Real Numbers

(ii) If (an) has a convergent subsequence with limit a then the sequence (an) itself will converge to a. Proof. Let (an) be Cauchy sequence and ? > 0 be given.

1 Review and Practice Problems for Exam # 2 - MATH 401/501 - Spring 2019

Instructor: C. Pereyra

Real numbers

Real numbers are closed under addition, multiplication, negation, subtraction and division by non-zero

real numbers. You are free to use usual arithmetic properties (commutative and associative properties

of addition and multiplication, distributive property, etc). Real numbers have an order, and obey a trichotomy ifx;yare real numbers then exactly one of the following holds:x=y,x < yorx > y.

Should know and be able to use

(i) the denition of absolute value of a real number, (ii) the triangle inequality (and \reverse" triangle inequality). Understand Archimedean properties and their implications: interspersing of integers byR, density of rationals and irrationals. Understand the meaning of upper and lower bounds for a set inR, and the meaning of the supremum (least upper bound or l.u.b.) and inmum (greatest lower bound or g.l.b.) of a set of real numbers. Be able to show that a given number is the supremum (inmum) of a set by showing that (i) it is an upper (lower) bound for the set, (ii) it is the smallest upper (largest lower) bound. Appreciate the Least Upper Bound and Greatest Lower Bound properties of real numbers: every non-empty and bounded set of real numbers has a unique supremum and a unique inmum.

Sequences of real numbers

Know the denition of bounded sequences, bounded above sequences and bounded below sequences. More precisely a sequencefxngn0is bounded (respectively bounded above or bounded below) i there isM >0 such that for alln0 we havejxnj M(respectivelyxnMorMxn). Know the;Ndenition of Cauchy sequences and of convergent sequences inRto a limitL2R. More precisely, a sequencefxngn0of real numbers {is Cauchy i given >0 there isN >0 such that for alln;mNthenjxnxmj , {converges toLi given >0 there isN >0 such that for allnNthenjxnLj .

Be able to show that limits are unique (that is if a sequence converges it converges to a unique limit).

Be able to prove or disprove that a given sequence converges or is Cauchy by using the \;Ndenition".

E.g.an= 1=n,bn= 2n.

Be able to show that a convergent sequence is a Cauchy sequence. Be able to show and use that Cauchy sequences (and hence convergent sequences) are bounded se- quences. However not all bounded sequences are convergent, e.g .bn= (1)nfor alln0. Be able to show that the sum/product of two Cauchy sequences (or two convergent sequences) is a Cauchy sequence (or a convergent sequence and convergent to the sum/product of the limits of the given convergent sequences \limit laws"). Understand that if a Cauchy (convergent) sequence is bounded away from zero then the sequence of

reciprocals is Cauchy (hence convergent and to the reciprocal of the limit which is necessarily non-zero,

another \limit law").

Be able to prove or disprove that a given sequence converges by appealing to additive/multiplicative/reciprocal

properties of limits (limit laws), and using known basic limits. 2 Know and be able to use the Monotone Bounded Sequence Convergence Theorem: (i) an increasing and bounded above sequence is convergent and to the sequence's supremum, (ii) a decreasing and bounded below sequence is convergent and to the sequence's inmum.

You should know and use some basic limits :

{limn!1xn= 0 ifjxj<1, is 1 ifx= 1, and does not exist ifx=1 orjxj>1; {limn!1x1=n= 1 ifx >0; { limn!11=n1=k= 0 for all integersk1. {limn!1n1=n= 1; { Letan>0,q2R, if limn!1an= 1>0 then limn!1aqn= 1. Appreciate the deep fact that Cauchy sequences are convergent sequences inR(completeness of the real numbers) .

Limit points, limsup, liminf

Appreciate the denition of \limit points" of a sequence as the collection of \subsequencial limits" (the

limits of convergent subsequences of the sequence. Know thatcis a limit point for a sequencefxngif for all >0 there are \innitely many" terms of the sequence in the interval [c;c+]. More precisely, for all;N >0 there is annNNsuch that jxnNcj (necessarily the set of labelsfnNgN0is an innite set!). Be able to identify the \limit points" (or \subsequencial limits") of a concrete sequence e.g:an= 3 for alln0,bn= (1)nfor alln0,cn= (1)nnfor alln0. Know that bounded sequences inRhave limit superior/inferior inR, dened as limsupfxng:= limN!1sup nNxnand liminffxng:= limN!1infnNxn. Be aware of the-Ncharacterization of limsup (similarly liminf): for all >0 (i) Finitely many terms of the sequencefxngare larger than limsupfxng+. More precisely for all >0 there isN >0 such that for allnNwe havexnlimsupfxng+. (ii) Innitely many terms of the sequencefxngare in between limsupfxng and limsupfxng+. More precisely, for all >0 andN >0 there isnNNsuch thatjxnNlimsupfxngj .

And its consequences:

{limsupfxngand liminffxngare limit points (subsequencial limits) of the sequence. {A sequence of real numbers converges if and only if the limsup and the liminf coincide. {The limsup is the \largest limit point" (or \largest subsequential limit") of the sequence, and liminf is the \smallest limit point" (or \smallest subsequential limit") of the sequence. {A sequence converges toLi all its subsequences converge toLi the unique limit point of the sequence isL. {Every bounded sequence has a at least one convergent subsequence or equivalently at least one \limit point" (Bolzano-Weierstrass theorem).

Be able to identify the limsup and liminf of a given sequence. Use this knowledge to conclude that if

limsupan= liminfan=Lthen the sequencefangconverges AND limn!1an=L. Be able to use the squeeze theorem to deduce convergence of the sequence being squeezed.

Series

Understand that convergence of a series is by denition convergence of the sequence of partial sums.

Be able to deduce from the theory of sequences basic convergence tests: Cauchy test, divergence test,

absolute convergence test, comparison test.

Be familiar with other useful tests such as: alternating series test, p-test, root test, and ratio test. Be

able to use these tests to deduce convergence or divergence of specic series. 3 Be able to exploit convergence properties of geometric series:1X n=0r nconverges to 1=(1r) ifjrj<1, diverges otherwise.

Limits and continuity of functionsf:E!R,ER

Know denition of a bounded function:9M >0 such thatjf(x)j Mfor allx2E. Know the equivalent \denitions" of limx!x0f(x) =L. LetERandf:E!R,x0is anadherent point

1ofE, then limx!x0;x2Ef(x) =Lif and only if

{(-denition)8 >09 >0 such thatjf(x)Lj 8x2Esuch thatjxx0j . {(Sequential denition) For all sequencesfxngn0inEif limn!1xn=x0then limn!1f(xn) =L. Know the equivalent \denitions" of continuity at a pointx0. LetERandf:E!R,x02E, thenfis continuous atx0if and only if {(Limit denition) limx!x0;x2Ef(x) =f(x0). {(-denition)8 >09 >0 such thatjf(x)f(x0)j 8x2Esuch thatjxx0j . {(Sequential denition) For all sequencesfxngn0inEif limn!1xn=x0then limn!1f(xn) =f(x0). Be able to decide whether a function is bounded or not and whether a function is continuous or not. Know that basic functions are continuous such us: constant function (f(x) =c), identity function (f(x) =x), absolute value function (f(x) =jxjforx2R), and exponential functions (f(x) =xpfor x >0, andg(x) =axfora >0 andx2R). Know the limit laws for functions and be able to prove them and use them to compute limits. Know and be able to prove that composition and arithmetic operations preserve continuity. Use these properties to conclude that more complex functions are continuous such us: polynomials (p(x) = a

0+a1x++anxn), rational functions (quotients of polynomials, wherever the denominator is

non-zero), exponentials with continuous base or exponent (f(x) =ap(x)org(x) =f(x)qwherefis a positive and continuous function andq2R. Use your knowledge of continuous functions to compute limits for example: {limx!x0xq=xq

0forx;x0>0 andq2R.

{limx!x0ax=ax0forx;x02Randa >0.

Practice Problems for Midterm #2

1. If the real numberxis not rational we sayxis \irrational".

(a) Show that ifp2Q,p6= 0, andxis irrational thenpxis irrational. (b) Show that ifx;y2Randx < ythen there is an irrational numberwsuch thatx < w < y (density of the irrational numbers).

2. For each subsetAof real numbers decide whether is bounded (above, below or both), nd supremum

and inmum: (a)A=f1;1=2;3g, (b)A=fn=(n+1) :n2N; n1g, (c)A=fr2Q:r <5g.

3. IfAandBare nonempty and bounded subsets ofRsuch thatABshow that inf(B)inf(A).

4. LetEbe a nonempty and bounded subset ofR, let2Rand >0. DeneE=fx:x2Ega

subset ofR. Prove that If0 then sup(E) =sup(E). What is inf(E)? What if <0?

5. Given >0 andfsngn0is a bounded sequence. Show that limsupfsng=limsupfsng. What can

you say when <0? (Hint use previous exercise).1

A pointx02ERis adherent i for all >0 there isx2Esuch thatjxx0j (in words, we can get arbitrarily close

tox0with pointsxinE). 4

6. For each of the following, prove or give a counterexample.

(a) Iffxngn0converges toxthenfjxnjgn0converges tojxj. (b) Iffjxnjgn0is convergent thenfxngn0is convergent .

7. We say the sequencefxngn0diverges to +1and we write limn!1xn= +1i for allM >0 there is

N >0 such that for allnNwe havexnM.

(a) Write down a denition for a sequencefyngn0to diverge to1. (b) Show that ifxnznfor alln0 andfxngn0diverges to +1thenfzngn0diverges to +1. (c) Letfxngn0sequence inR,xn>0. Show that limn!1xn= +1if and only if limn!1(1=xn) = 0.

8. The sequence of positive real numbersftngn0converges tot. Decide whether the following sequences

are convergent or not. If convergent explain why and identify the limit, if not convergent explain why.

Find the limsup and liminf of each sequence.

(a)an=pt n, (b)bn= 5t3nt2n+ 7, (c)cn=n2 n(1)n, (c)dn=n+tn.

9. Use squeeze theorem and properties of sine function to show lim

n!1sinnn = 0.

10. Show that the sequence dened byx1= 1 andxn+1=p1 +xnforn1 is convergent (hint: show

that it is increasing and bounded by 2). Find the limit.

11. Letxn=nsin2(n=2). Find the setSof limit points (subsequencial limits), nd limsupxnand

liminfxn. (Assume known properties about sine function.)

12. Show that the sequence of partial sums:Sn= 1 +12

+13 ++1n dened forn1 is not Cauchy (hint: show thatS2nSn1=2). Conclude that the harmonic series is divergent.

13. A sequencefsngn0is contractive if there is a constantrwith 0< r <1 such thatjsn+2sn+1j

rjsn+1snjfor alln0. Show that a contractive sequence is a Cauchy sequence and hence a convergent sequence (hint: recall convergent geometric series).

14. Show that if a series converges absolutely then it converges.

15. Assume thatjanj 2bn+ 3nfor alln0 and1X

n=0b nconverges. Show that1X n=0a nconverges.

16. Determine for eachx2Rwhether the series1X

n=12 nxnn is convergent or divergent.

17. Letan>0 for alln1. Show that liminfa1=nnliminfan+1a

18. LetER,f;g:E!Rbe functions,x0an adherent point ofE. Assumefhas limitLatx0inE

andghas limitMatx0inE. Show that limx!x0;x2Ef(x)g(x) =LM. Deduce that the product of two continuous functions atx0is continuous atx0.

19. Show that the functionf:R!Rdened to bef(x) = 0 ifx2Qandf(x) =1 ifx =2Qis nowhere

continuous.

20. Letp2R. Show that the functionf: (0;1)!Rgiven byf(x) =xpis continuous on (0;1). Hint:

use that wheneveran>0 and limn!1an= 1 then limn!1(an)q= 1:

21. Show that the functionf(x) =jxjis continuous onR.

22. Study the continuity properties of the functionf: [1;1]!Rgiven byf(x) =x2if1x <0

x+ 1 if 0x1:quotesdbs_dbs6.pdfusesText_11

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