推 herstein:如果沒有用道|f_n'|<=2, 我給你一個反例f_n(x)=x^n [0,1] 03/16 15:39
有勞各位先進了
{fn:R to R} function sequences such that fn is continuous for all n, x
and fn(0)=0 for all n, |fn'(x)|≦2 for all n, x
If lim fn(x)=g(x) for all x
n
show that g is continuous.
我的作法是這樣子
fix a
For any n
since fn is conti. at a
exist δ depends on n and a s.t |fn(x)-fn(a)|<ε/3 if |x-a|<δ
for any x in (a-δ, a+δ) since lim fn(x)=g(x) for all x
there is
n1 s.t |fn(x)-g(x)| <ε/3 if n>n1
n2 s.t |fn(a)-g(a)| <ε/3 if n>n2
choose n>n1 and n>n2
then |g(x)-g(a)|<|g(x)-fn(x)|+|fn(x)-fn(a)|+|fn(a)-g(a)|<ε
請問這樣子對嗎？因為我沒有用到題目給的條件，覺得怪怪的！
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