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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 11 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

MATH 401 NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously7 we have studied sequences of real numbers Now we discuss the topic of sequences of real valued functions A sequence of functions fn is a list of functions f17f27 such that each fn maps a given subset D of R into R I Pointwise convergence De nition Let D be a subset ofR and let fn be a sequence offunctions de ned on D We say that fn converges pointwise on D if lim fnx exists for each point z in D Hoe This means that lim fnx is a real number that depends only on x Hoe lf fn is pointwise convergent then the function de ned by u lim fnx7 for every x in D7 is called the pointwise limit of the sequence 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 for any x gt 0 Example 2 Let fn be the sequence of functions on R de ned by fnx This sequence converges pointwise to the zero function on R Example 3 Consider the sequence fn offunctions de ned by 2 fnx w for all z in R n Show that fn converges pointwise Solution For every real number d we have naoo n 2 l l limfnxlim x lime 2 limi 000 aw n2 naoo 71 new n2 1 Thus7 fn converges pointwise to the zero function on R Example 4 Consider the sequence fn offunctions de ned by i sinnz 3 7 Mi 1 Show that fn converges pointwise for all z in R M96 Solution For every x in R7 we have 71 S s1nnz3 lt 1 n1 n1 i n1 Moreover7 lim 7 Hoe n 1 Applying the squeeze theorem for sequences7 we obtain that lim fnz 0 for all z in R Therefore7 fn converges pointwise to the function f E 0 on R Example 5 Consider the sequence fn of functions de ned by fnx 71 for 0 S x S 1 Determine whether fn is pointwise convergent Solution First of all7 observe that fn0 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 m But ln lt 0 when 0 lt z lt 17 it follows that lim fnz0 for0ltxlt1 Finally7 fn1 n2 for all n So7 lim fn1 oo Therefore7 fn is not pointwise convergent on 01 Example 6 Let fn be the sequence offunctions de ned by fnx cos z 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 2 It follows that Jirnocosz 0 for z 31 0 Moreover7 since fn0 1 for all n in N one gets lirn fn0 1 Therefore7 fn converges pointwise to the function f de nedlby0 0 if 71ltdlt0 or 0ltdltI f1 if 2350 72 Example 7 Consider the sequence offunctz39ons de ned by fnx nd17 x on 01 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 nln17m nln17m 0 lirn fnx lirn nze z lirn ne Hoe Hoe Hoe because ln1 7 d lt 0 when 0 lt z lt 1 Therefore7 the given sequence con verges pointwise to zero Example 8 Let fn be the sequence offunctz39ons on R de ned by n3 if 0 lt x S l fnw T 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 07 171 get smaller as n 7 oo Therefore7 fnx 1 for all n gt N Hence7 lirn fnx 1 for all x The formal de nition of pointwise convergence Let D be a subset of R and let fn be a sequence of real valued functions de ned on D Then fn converges pointwise to f if given any x in D and given any 8 gt 07 there exists a natural number N Nx7 8 such that lfnx 7 lt 8 for every n gt N Note The notation N Nz7 8 means that the natural number N depends on the choice of z and 8 I 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 8 gt 07 there exists a natural number N N8 such that lfnz 7 lt 8 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 implies pointwise convergence But the con verse is false as we can see from the following counter example Example 9 Let fn be the sequence offunctions on 07 00 de ned by in 1 nzzz39 M96 This function converges pointwise to zero lndeed7 1 nzxz nzsz as n gets larger and larger So7 i m 1 1 grif n 31 gin 0 But for any 8 lt 127 we have fifi 0gt 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 f is continuous on D Homework Problem 1 Let fn be the sequence of functions on 07 1 de ned by fnx nx17x4 Show that fn converges pointwise Find its pointwise limit Problem 2 1 ls the sequence of functions on 07 1 de ned by fnz 1 7 z pointwise convergent Justify your answer Problem 3 Consider the sequence fn of functions de ned by fnz 712 for all z in R Show that fn is pointwise convergent Find its pointwise limit Problem 4 Consider the sequence fn of functions de ned on 07 7T by fnx sin Show that fn converges pointwise Find its pointwise limit Using the above theorem7 show that fn is not uniformly convergent

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