逐点收敛与一致收敛.pdf

MATH 401 NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously 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 f1 f2 such that each fnmaps a given subset D of R into R I Pointwise convergence Defi nition Let D be a subset of R and let fn be a sequence of functions defi ned on D We say that fn converges pointwise on D if lim n fn x exists for each point x in D This means that lim n fn x is a real number that depends only on x If fn is pointwise convergent then the function defi ned by f x lim n fn x for every x in D is called the pointwise limit of the sequence fn Example 1 Let fn be the sequence of functions on R defi ned by fn x nx This sequence does not converge pointwise on R because lim n fn x for any x 0 Example 2 Let fn be the sequence of functions on R defi ned by fn x x n This sequence converges pointwise to the zero function on R Example 3 Consider the sequence fn of functions defi ned by fn x nx x2 n2 for all x in R Show that fn converges pointwise Solution For every real number x we have lim n fn x lim n x n x2 n2 x lim n 1 n x2 lim n 1 n2 0 0 0 1 Thus fn converges pointwise to the zero function on R Example 4 Consider the sequence fn of functions defi ned by fn x sin nx 3 n 1for all x in R Show that fn converges pointwise Solution For every x in R we have 1 n 1 sin nx 3 n 1 1 n 1 Moreover lim n 1 n 1 0 Applying the squeeze theorem for sequences we obtain that lim n fn x 0for all x in R Therefore fn converges pointwise to the function f 0 on R Example 5 Consider the sequence fn of functions defi ned by fn x n2xnfor 0 x 1 Determine whether fn is pointwise convergent Solution First of all observe that fn 0 0 for every n in N So the sequence fn 0 is constant and converges to zero Now suppose 0 x 1 then n2xn n2enln x But ln x 0 when 0 x 1 it follows that lim n fn x 0for 0 x 1 Finally fn 1 n2for all n So lim n fn 1 Therefore fn is not pointwise convergent on 0 1 Example 6 Let fn be the sequence of functions defi ned by fn x cosn x for 2 x 2 Discuss the pointwise convergence of the sequence Solution For 2 x 0 and for 0 x 2 we have 0 cos x 1 2 It follows that lim n cos x n 0for x 6 0 Moreover since fn 0 1 for all n in N one gets lim n fn 0 1 Therefore fn converges pointwise to the function f defi ned by f x 0if 2 x 0or0 x 2 1ifx 0 Example 7 Consider the sequence of functions defi ned by fn x nx 1 x non 0 1 Show that fn converges pointwise to the zero function Solution Note that fn 0 fn 1 0 for all n N Now suppose 0 x 1 then lim n fn x lim n nxenln 1 x x lim n nenln 1 x 0 because ln 1 x 0 when 0 x 1 Therefore the given sequence con verges pointwise to zero Example 8 Let fn be the sequence of functions on R defi ned by fn x n3if0 N Hence lim n fn x 1for all x The formal defi nition of pointwise convergence Let D be a subset of R and let fn be a sequence of real valued functions defi ned on D Then fn converges pointwise to f if given any x in D and given any 0 there exists a natural number N N x such that fn x f x N Note The notation N N x means that the natural number N depends on the choice of x and 3 I Uniform convergence Defi nition Let D be a subset of R and let fn be a sequence of real valued functions defi ned on D Then fn converges uniformly to f if given any 0 there exists a natural number N N such that fn x f x Nand for every x in D Note In the above defi nition the natural number N depends only on Therefore 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 of functions on 0 defi ned by fn x nx 1 n2x2 This function converges pointwise to zero Indeed 1 n2x2 n2x2as n gets larger and larger So lim n fn x lim n nx n2x2 1 x lim n 1 n 0 But for any Hence fn is not uniformly convergent Theorem Let D be a subset of R and let fn be a sequence of continuous functions on D which converges uniformly to f on D Then f is continuous on D 4 Homework Problem 1 Let fn be the sequence of functions on 0 1 defi ned by fn x nx 1 x4 n Show that fn converges pointwise Find its pointwise limit Problem 2 Is the sequence of functions on 0 1 defi ned by fn x 1 x 1 npointwise convergent Justify your answer Problem 3 Consider the sequence fn of functions defi ned by fn x n cos nx 2n 1 for all x in R Show that fn is pointwise conv