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N cxg tf. ۃX c ( X e b v O t)RS42PS A C X ` g Z p \ R f X N 390X390X4mm I W iJAN R h j ̃y W ł B i 4 `5 c Ɠ ȓ ɔ ܂ i y j j B DCM I C ̓A C X ` g Z( ) ̃p \ R f X N w z Z ^ ʔ̃T C g ł BDCM I C ł͉Ƌ E C e A ͂ ߂Ƃ A 34 _ ̏ i 舵 Ă ܂ B z Z ^ ʔ DCM I C ł̂ y ݂ B. 6 Suppose that (fn) is a sequence of continuous functions fn R → R, and(xn) is a sequence in R such that xn → 0 as n → ∞Prove or disprove the following statements (a) If fn → f uniformly on R, then fn(xn) → f(0) as n → ∞ (b) If fn → f pointwise on R, then fn(xn) → f(0) as n → ∞ Solution • (a) This statement is true To prove it, we first observe that f is con. R W A A c w Z E ̉ t U F M ^ N G C ^ ̏ڍׂɂ Čf ڂ Ă ܂ B( { s k ) ڂ̃J L ̏Љ Ă ܂ ̂ŁA i H I т̎Q l ɂ Ă݂Ă B.

()i t n i t n N n N n N n N n N n A e n B e n q t N q t q t q t Ω −Ω = 1 1 1 1 3 2 1 sin sin 3 sin 2 sin 1 π π π π M M, (7) where Ω =2~sinn 2()N 1 n ω π As we mentioned last time, these modes are essentially standing waves Let's see that this is the case by writing Eq. W ed n esd ays F r i d ays 215p m 515p m T h e Ma y c r o ft 1 4 7 4 C o l u m b i a R d N W T h e Maycr o ft 1474 Co l u mb i a Rd NW T h e Ma y c r o ft 1 4 7 4 C o l u mb i a R d N W T h e Maycro ft 1474 Co l u mb i a Rd NW S atu r d ays 100p m 330p m Mar th a' s O u tfi tter ' s 24 Mar ti n L u th er Ki n g Jr Ave S E. Z N g V b v ̘V ܁ASHIPS( V b v X) ̌ T C g E c I C V b v BErrico Formicola N V b N p ^ X g C v l N ^ C X ܏ 񂩂 X ^ C O āA V A C e ̂ Љ ȂǖL x ȃR e c p ӁBWeb ʔ̂ X ܂Ƌ ʃ C g ,000 ~ ȏ ő B ŒZ B.

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T i=1 2=n= p 2 = Hence his a continuous function from Rminto Rn 4010 Let fa ng2l1 Prove that fde ned by f(fb ng) = P 1 n=1 a nb n is a continuous realvalued function on l1 2 Solution Let fb ng2l1 and >0 Since fa ng2l1there exists some M such that ja nj. G g ̃ ^ T C g A g g ̗ s A A w A z A w w Z A d A s ̏ 񂩂 A I X X ̂ X A X g ܂ŏڂ Љ ܂ B f ŁA N V t @ C h ܂ B g g E A W A ۉf 10 N11 9 15 ɊJ A A W A n f 搻 ҂ɂ 錻 f ̉f ՂƂ ăJ i _ ōŒ ̉f ՁB { f Q f Ղ̏ڍׂ Љ B. R ` t @ N g I C X g A.