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# Introduction to Analysis I MATH 401

Penn State

GPA 3.88

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This 0 page Class Notes was uploaded by Elaina Osinski on Sunday November 1, 2015. The Class Notes belongs to MATH 401 at Pennsylvania State University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/233001/math-401-pennsylvania-state-university in Mathematics (M) at Pennsylvania State University.

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Date Created: 11/01/15

SPRING 2009 MATH 401 NOTES Sequences of functions Pointwise and Uniform Convergence Previously7 we have studied sequences of real numbers Now we discuss sequences of real valued functions By a sequence fn of real valued func tions on D7 we mean a sequence f17f27 7fn7 such that each fn is a function having domain D and range a subset of R I Pointwise convergence De nition Let D be a subset ofR and let fn be a sequence offunctz39ons de ned on D We say that fn converges pointh39se on D if lim fnz em39sts for each point z in D Hoe In other words7 lim fnx must be a real number that depends only on x Hoe In this case7 we write We hm Me VLHOO for every x in D and f is called the pointh39se limit of the sequence Formal De nition The sequence fn converges pointwise to f on D if for every x E D and for every 6 gt 07 there exists a natural number N Nx7 6 such that lfnz 7 lt 6 whenever n gt N Note The notation N N78 means that the natural number N depends on the choice of z and 8 Example 1 Let fn be the sequence of functions on R de ned by fnz nx This sequence does not converge pointwise on R because lim fnx 00 Hoe for any x gt 0 Example 2 Let fn be the sequence of functions on R de ned by This sequence converges pointwise to the zero function on R lndeed given any 6 gt 0 choose N gt f then fn0 ltlt6 forngtN Example 3 Consider the sequence fn offunctions de ned by Mn 7 kW for all z in R Show that fn converges pointwise Solution For every real number d we have 2 2 1 1 hm Mm hm 1 1 x2 lirn 72z hm 71 001 1 naoo naoo n n naoo n naoo n Thus fn converges pointwise to the function u 1 on R Example 4 Consider the sequence fn of functions de ned by fx 71 for 0 S x S 1 Determine whether fn is pointwise convergent on 01 Solution First of all we observe that f0 0 for every n in N So the sequence fn0 is constant and converges to zero Now suppose 0 lt z lt 1 then 71 nZenh W a 0 as n a 00 Finally f1 n2 for all n So lirn f1 00 Therefore fn is not pointwise convergent on 01 Al though it is pointwise convergent on 01 Example 5 Consider the sequence fn offunctions de ned by 7 sinnz 3 7 n 1 Show that fn converges pointwise for all z in R M96 Solution For every x in R we have 71 S s1nnz3 lt 1 n1 n1 i n1 Moreover lirn 7 Hoe n1 Applying the sandwich theorem for sequences7 we obtain that lim fnz 0 for all z in R Hoe Therefore7 fn converges pointwise to the function f 0 on R Example 6 Let fn be the sequence offunctions de ned by fnx cos x for 7T2 S x S 7r2 Discuss the pointwise convergence of the sequence Solution For 7T2 S x lt 0 and for 0 lt x S 7r27 we have 0 S cosx lt 1 It follows that lim cosz 0 for z 31 0 Moreover7 since fn0 1 for all n in N7 one gets lim fn0 1 Therefore7 fn converges pointwise to the function f de ned by I 2 0 if 71ltlt0 or 0ltlt 27 i fa 1 if 950 Example 7 Consider the sequence fn offunctions de ned by W W Show that fn converges pointwise for all z in R Solution Moreover7 for every real number d we have i i d 3320an i 313 W Hence7 fn converges pointwise to the zero function Example 8 Consider the sequence offunctions de ned by fnx n17 x on 071 Show that fn converges pointwise to the zero function Solution Note that fn0 fn1 07 for all n E N Now suppose 0 lt z lt 17 then lirn fnz 0 Therefore7 the given sequence converges pointwise to zero Example 9 Let fn be the sequence offunctz39ons on R de ned by n3 if 0 lt x S l fnw 7 1 otherwise Show that fn converges pointwise to the constant function f 1 on R Solution For any x in R there is a natural number N such that z does not belong to the interval 07 1N The intervals 07 1n get smaller as n a 00 We see that fnx 1 for all n gt N Hence7 lirn fnx 1 for all x II Uniform convergence De nition Let D be a subset of R and let fn be a sequence of real valued functions de ned on D Then fn converges uniformly to f if given any 6 gt 07 there exists a natural number N Ne such that Una 7 lt E for every n gt N and for every x in D Note In the above de nition the natural number N depends only on 8 Therefore7 uniform convergence irnplies pointwise convergence But the con verse is false as we can see from the following counter example Example 10 Let fn be the sequence offunctz39ons on 07 00 de ned by nx 1 nzzz39 M96 This sequence converges pointwise to zero lndeed7 1 nzzz 71de as n gets larger and larger So7 71 1 1 law 31330 31330 0 4 But for any 8 lt 127 we have fifii0gt n n 2 Hence fn is not uniformly convergent Theorem Let D be a subset ofR and let fn be a sequence ofcontinuous functions on D which converges uniformly to f on D Then its limit f is continuous on D Example 10 Let fn be the sequence of functions de ned by fnx cos x for 7T2 S x S 7r2 Discuss the uniform convergence of the se quence Solution We know that fn converges pointwise to the function f de ned by see Example 6 I 2 0 if 71ltlt0 or 0ltlt 7 27 7 f 1 if 950 Each fnz cos z is continuous on 771727 7r2 But the pointwise limit is not continuous at z 0 By the above theorem7 we conclude that fn does not converge uniformly on 771727 7r2 Example 11 Consider the sequence fn offunctions de ned by i sinnz 3 Prove that fn converges uniformly to the zero function on R Solution We have seen that fn converges pointwise to the zero function on R see Example 5 Moreover for all z in R lsinn 3 lt 1 W 7 n 139 Given any 6 gt7 we can nd N E N such that 1 M It follows 7 lt E for every n gt N and for every x inR Hence7 fn converges uniformly to the zero function on R moo Ol lt 6 whenever n gt N

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