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Bounded Sets Sequences Limits of Sequences Rules for Limits Math 361 Basic facts from Math 360 August 14 2003 A subset S of the real numbers is bounded from below if there is some number In so that m S x Vac E S and bounded from above if there is a number M such that M 2 x Vac E S If a set is bounded from above and below then we say it is bounded If a set is bounded from below then we de ne infS as the least upper bound of the set m If S is bounded from above we de ne supS to be the greatest lower bound of the set meVx Si M2 xSMVx Si The most important idea covered in math 360 was the concept of a sequence A sequence is a function from the positive integers natural numbers N to some set The simplest examples aroe sequences of real numbers R Using standard functional notation we could denote the function as xNgtR then the nth term of the sequence would be denoted It is customary not to use functional notational but rather to denote the use subscripts so the nth term is denoted by as Almost as important as the concept of a sequence is the concept of a subsequence Given a sequence 9 we de ne a subsequence by selecting a subset of and keeping them in the same order as they appear in In practice this amounts to de ning a function from N to itself We denote this function by nj It must have the following property nj lt nj1i The j h term of our subsequence is given by 957 As an example consider the sequence 95 71 71 the mapping nj 2quot de nes the subseqence yon il2j2ji A sequence of real numbers has a limit if there is a number L such that given any 6 gt 0 there exists a N gt 0 such that in 7 L lt 6 whenever n gt N A sequence with a limit is called a convergent sequence The limit when it exists is unique A sequence may itself fail to have limit but it may have a subsequence which does In this case the sequence is said to have a convergent subsequence For example 957 71 is not convergent but the subsequence de ned by nj 2quot is We have several rules for computing algebraic combinations of convergent se quences Typeset by AMSTEX Algebraic Rules for Limits Suppose that x are convergent sequences of real numbers then lim 7x7 exists and equals 7 lim x naoo W400 limnnodx y exists and equals lim xn lim yn ngtoo ngtoo limnnodxny exists and equals lim lim y ngtoo ngtoo x lim x limngoo n exists provided hm yn y 0 and equals yn new 11 Moo yn In this theorem the nonitrivial claim is that the limits exist once this is clear it is easy to show what they must be BolzanoWeierstrass A problem of fundamental importance is to decide whether or not a sequence has a limit A sequence is bounded if there is a number M such that lx l lt M for all n A sequence is noniincreasing if xn 2 xn1 for all n and nonidecreasing if xn S xn1 for all n The completeness axiom of the real numbers states Completeness Axiom A bounded noniincreasing or nonidecreasing sequence has a limit If a bounded sequence is neither nonidecreasing nor noniincreasing then the only general theorem about convergence is BolzanoiWeierstrass Theorem A bounded sequence of real numbers always has a con vergent subsequence Note that this does not assert that any bounded sequence converges but only that any bounded sequence has some subsequence which converges In general if S C R then the set of points which can be obtained as limits of sequences x C S is called the set of accumulation points of St A subset S is said to be dense in an interval I whenever I is a subset of the set of accumulation points of St For example the rational numbers Q are dense in every intervals The following two lemmas are very useful Lemma 1 If x yn 2n are sequences of real numbers such that In S yn S Zn and xn and Zn are convergent with L lim xn lim 2n ngtoo ngtoo then yn converges with lim yn Li ngtoo Lemma 2 If xn 2 0 is convergent then lim xn 2 0i ngtoo Cauchy Sequences In the above discussion of limits we always assumed that the limiting value is known in advance There is a criterion due to Cauchy which implies that a given sequence has a limit which makes no reference to its va uei 2 Cauchy Criterion for Sequences If is a sequence of real numbers such that given 6 gt 0 there exists an N for which lacn 7 xml lt 6 whenever both it and m are greater than N then the sequence is convergent Series A series is the sum of a sequence This is usually denoted by 00 E 9 n1 A series converges if the sequence of partial sums k 3k g xn n1 converges In this case the sum of the series is de ned by oo 2 lim 3k kaoo n1 If a series does not converge then it diverges A series converges absolutely if the sum of the absolute values 00 n1 converges The following theorem describes the elementary properties of series Theorem on Series Suppose that 9cm yn are sequences Suppose that E 9cm 2 y converge t en Emmi 371 converges and yn Z Zym fa E R Zoos ame fac 2 0 for all n then 2 2 0 Convergence Tests There are many criteria that are used to determine if a given series converges The most important is the comparison test Comparison Test Suppose that 9cm y are sequences such that S y y converges then so does IfO S y S acn and By diverges then so does To apply this test we need to have examples of series which we know are convergent or divergent The simplest case is a geometric series This is because we have a formula for the partial sums From this formula we immediately conclude Convergence of Geometric Series A geometric converges if and only if lal lt 1 The root and ratio tests are really special cases of the comparison test where the series are comparable to geometric series Ratio Test If an is a sequence with xn 1 hm sup ngtocgt n then the series 00 Z converges if oz lt 1 xn n1 diverges if oz gt 1 The test gives not information if oz 1 We also have Root Test If an is a sequence with 1 llmsup lacle oz ngtocgt then the series i0 converges if oz lt 1 x n1 n diverges oz gt 1 The test gives not information if oz 1 If oz lt 1 in the ratio or root tests then the series converge absolutely Another test is obtained by comparing a series to an integra A Convergence Test If acn is a monotone decreasing positive sequence then 00 00 Zorn converges and only if Z2kl 2k n1 1c1 Using this test we can easily show that the sum M8 1 H converges if and only ifp gt 1 A nal test which is sometimes for showing that a series with terms that alternate in sign converges is Alternating Series Test Suppose that an is a sequence such that the sign alternates the limnH00 x7 0 and lacn1l S then oo 2 n1 converges Note that this test requires that the signs alternate the absolute value of the sequence is monotonely decreasing and the sequence tends to zero If any of these conditions are not met the series may fail to converge This test is a consequence of a very useful formula the partial summation formu a M3 m 1an 7 257017 7 an am1Bm 1 n1 H where B7 Z bl 11 4 Limits of Functions The next thing to consider is the behavior of functions de ned on intervals in R Suppose that is de ned for x E a c U c b This is called a punctured neighborhood of c We say that of has a limit L as x approaches c if given 6 gt 0 there exists 6 gt 0 such that Ll lt 6 provided 0 lt lx7 cl lt 6 and we write lim L Note that in this de nition nothing is said about the value of at x c This has no bearing at all on whether the limit exists lf is de ned and we have that gig m 1 then we say that is continuous at x c If is continuous for all x E a b then we say that is continuous on a b In addition to the ordinary limit we also de ne one sided limits lf is de ned in a b and there exists an L such that given 6 gt 0 there exists 6 such that 7 Ll lt 6 provided 0 lt x 7 a lt 6 then limJr L If instead 7 Ll lt 6 provided 0 lt b7 x lt 6 then llIIbL L The rules for dealing with limits of functions are very similar to the rules for handling limits of sequences Algebraic Rules for Limits of Functions Suppose that fxgx are de ned in a punctured neighborhood ofc and that lim L limgx M Then lim7fx exists and equals 7 L limfx exists and equals L M limfxgx exists and equals LM lim M exists p39rouided M y 0 and equals H gm M From this we deduce the following results about continuous functions Algebraic Rules for Continuous Functions Iffx gx are continuous at x c then so are 7fx fxgx Ifgc y 0 then is also continuous atx c For functions we have one further operation which is very important composition Continuity of Compositions Suppose that fxgy are two functions such that fx is continuous at x c and gy is continuous at y then the composite g o is continuous at x c Uniform Continuity A function de ned on an interval la bl is said to be uniformly continuous if given 6 gt 0 there exists 6 such that lf1fyllt sway E aabl with lxiyl lt 5 The basic proposition is Proposition A continuous function on a closed bounded interval is uniformly continuous Using similar arguments we can also prove MaxEMin theorem for Continuous Functions If x is continuous on a closed bounded interval a b then there exists an E a b and 952 E a b which satisfy an sup x m2 inf x cEab Eglavbl As a nal result on continuous functions we have lntermediate Value Theorem Intermediate Value Theorem Suppose thatfx is continuous on 1 b and u lt fb then given y E fafb there exists c E a b such that y Differentiability A function de ned in a neighborhood of a point c is said to be differentiable at c i the function f 96 i f C 995 7 de ned in a deleted neighborhood of c has a limit as x A c This limit is called the derivative of f at c we denote it by f A function which is differentiable at every point of an interval is said to be differentiable in the interval If the derivative is itself continuous then the function is said to be continuously differentiable As with continuous functions we have algebraic rules for differentiation Rules for Differentiation Suppose that x 9a are di erentiable at x c then so are Efx If gc y 0 then so is The derivatives are given by f c fCDa f 9 Cf c 9 f9 c f c9c fcgCa f f 7 f C9C 7 fc9 c a C gltcgt2 In addition we can also differentiate a composition The Chain Rule If is di erentiable at ac c and gy is di erentiable at y then 9 o is di erentiable at ac c the derivative is g 0 HQ 9 fcf c Little 0 and big 0 It is often useful to be able to compare the size of two functions 9a near a point ac c without being too speci c When we write ogx near to ac c it means that lim we 996 If we write the Ogx near to ac c this means that we can nd an M and an e gt 0 such that Mac 996 lt Mgx provided lac 7 cl lt e ie limsup lt 0 6 For example a function is differentiable at ac c if and only if there exists a number L for which ffx fC Lx7 c 0lx7 cl Of course L fc Higher Order Derivatives If the derivative of function happens itself to be differentiable then we say that is twice differentiable The second derivative is denoted by f aci ln ductively if the km derivative happens to be differentiable then we say that f is k 17times differentiable We denote the km derivative by flkl Taylor s Theorem For a function which has n derivatives we can nd a polynomial which agrees with to order n 7 1 at a point Taylor s Theorem Suppose that has n derivatives at a point c then n71 v v fm c ac 7 c J M 7 Z Rm 10 j where RWU OW Cln There are many different formulae for the error term One such expression is aE xcifacltc aE cxifacgtc n a Rnx f 7 c where n An important special case of Taylor s Theorem is the mean value theorem Mean Value Theorem Suppose that is continuous on a b and di erentiable on a b then there exists a c E a b such that N ll fCW a Open and Closed sets A subset S of R is said to be open if for each ac E S there is an e gt 0 such that x7exe C St A set S is said to be closed if its complement SC R S is open Properties of open sets If SEX at E A is a collection of open set then U SEX is open i CXEA HA is a nite set then m SEX is open i EXEA Using De Morgan s laws A0 BC ACU BC AU BC Ac BC one easily derives Properties of closed sets If SEX at E A is a collection of closed set then m SEX is open ozEA HA is a nite set then U SEX is open as If S C R then we de ne the set of limit points of S by Sac foranyegt0xiexe Sx Characterization of closed sets by limit points A subset S C R is closed if and only if S C S For an arbitrary subset we de ne the closure of S by cl S S S U S This is always a closed set We also de ne the interior of S by intA U A AcsAopen Compact sets A set S C R is compact if the following property holds Every sequence C S has a limit point in S HeineiBorel Theorem A set S is compact if and only if it is closed and bounded A collection of sets B Ba oz E A de ne a cover of a set S provided SC U Ba as If each Ba is open set then we say that B de nes an open cover of S If there is a nite subset 11 am C A such that SCBalUnUBam then we say that B contains a nite subcover We have the following alternative character ization of compact sets Open cover de nition of compactness A subset S C R is compact if and only if every open coue39r ofS contains a nite subcoue39r One of the reasons compactness is an important property is the following Continuous functions on compact sets If f is a continuous function de ned on the compact set S then is compact This result has a very important corollary Corollary A continuous function assumes its minimum and maximum values on a com pact set Connected sets A subset S of R is said to be disconnected if there exists two open sets A B such that AnBZJ SCAUB Ansy andany Q The only connected subsets of R are rays and intervals a b a b a b a b Let S C R and suppose that A C S has the following properties A is connected If B is a connected subset of S then either B C A or A 0 B ll We say that A is a connected component of S The connected components of a set are its largest connected subsets One can show that the connected components of an open sets are all of the form a b and establish the following Structure of open sets IfS is an open subset ofR then there is a countable collection of disjoint intervals aj bj such that S U 0391 51 j1 These intervals are the connected components of S One of the reasons connectedness is an important property is the following Continuous functions on connected sets If f is a continuous function de ned on the connected set S then is connected Power Series A very special type of in nite series is a power series 00 E andquot n0 We de ne the number 1 p limsup la l ngtocgt Convergence for power series pr 00 then the power series diverges for all x y 0 If p lt 0 then the power series converges absolutely for as C 71 717 and diverges for plxl gt 1 In fact one can show that in case p lt 0 then the power series is an in nitely di erentiable function of as C L 1 Many other convergence criteria for power series can be obtained from the convergence criteria for numerical series given above