※ 引述《wyob (Go Dolphins)》之铭言： : f_n(x)=(nx)/(1+nx^2) : 请问x在[0,1]和在[1/π,2]这两个区间内都是均匀收敛吗?? : 常看到这些题目，所以把它弄懂 : 感谢 0 , x=0 lim f_n(x) = n→inf 1/x , x€(0,inf) (pf：1.x=0： trivial 2.x€(0,inf)： nx x (nx)/(1+nx^2)=─────=───── 1+nx^2 1 + x^2 ─ n when n goes to inf, the limit will be x/x^2 = 1/x) Problem 1. x€[0,1] We use the thereom： At (C[a,b], supnorm) if 1.fn(x) € (C[a,b], supnorm) oo 2.(fn(x)) are uniformly Cauchy in (C[a,b], supnorm) n=1 then by Denoting lim fn(x) = f(x) (because in (C[a,b], supnorm) will be complete) n→oo we have 4.fn(x) converges to f(x) uniformly 5.f(x) € (C[a,b], supnorm) --------------------------- Assume fn(x) satisfies 2. Since fn(x) satisfies 1. By the theorem we have 5., leading to contradiction ( f(x) is not continuous at 0) so fn(x) is not uni. Cauchy absolutely not uni. conv. Problem 2. x€[1/pi,2] Consider │fn(x)-f(x)│ =│ nx 1 │ │─── - ─ │ 1+nx^2 x 1 1 =│────│ <= │──────│---* x(1+nx^2) π(1+nπ^2) Since * conv. to 0 for all epsilon .......................blablabla Done~ --

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