Anonymous

# 高等微積分^*^

Prove: If Σ(n=1→ ∞) u_n(x) is uniformly convergent for a<=x<=b, then the series is uniformly convergent in each smaller interval contained in the interval a<=x<=b.More generally, if a series is uniformly convergent for a given set E of values of x , then it is uniformly convergent for any set E_1 that is part of E. (過程請詳答)

Rating

由定義即得!

2009-08-31 22:53:12 補充：

已知Σ[n=1~∞] u_n(x) = f(x) uniformly in E

則For all ε>0, there exists N>0 (N is indep. on x), such that

|Σ[n=N~∞] u_n(x) - f(x) |<ε, for all x in E,

thus

|Σ[n=N~∞] u_n(x) - f(x)|<ε, for all x in E_1.

由紅字部分知 Σ[n=1~∞] u_n(x) = f(x) uniformly in E_1