高等微積分*-^

任意 ε>0 ,因Σ[n=1~∞] M_n 收斂, 故存在N>0,使m>n>N時
|Σ[k=n~m-1] M_k |<ε,
又| f_m(x)-f_n(x) | = | f_m(x)-f_m-1(x) |+...+| f_n+1(x)-f_n(x) |
<= M_m-1 +....+ M_n <ε (與 x無關, 只要m,n>N就可以)
則數列 {f_n(x)}為Cauchy 數列
故數列 {f_n(x)} conv. uniformly.

Σ(n=1→ ∞) |f_n(x)| < Σ(n=1→ ∞) M_n - f_1(x) < ∞,
by Weierstrass-M test. f_n(x) is uniform convergent.

兩題都用Cauchy sequence檢驗法即得!