※ 引述《charliehope (○雲淡風清○隨緣～)》之銘言：
: 不知有沒有辦法證明
: if fn is a sequence of functions s.t
: fn are all bounded, concave, continuous
: and fn → f uniformly
: then f is bounded..
: 還是說f有可能不是bounded??
: 先感謝各位答覆
條件只要：
f_n：S→R , f：S→R
1.f_n is bdd. for all n
2.f_n → f uniformly
then f is bdd. on S
pf：
Since f_n is bdd. for all n
│f_n(x)│<= M_n , for all x€S
and since f_n → f uniformly
take ε=1 , there exists N>0 s.t.│f_n(x)-f(x)│< 1 , for all n>= N
take n=N , by triangle ineqality , │f(x)│< 1 +│f_N(x)│ <= 1 + M_N
Hence f(x) is bdd. on S
除此之外 由此亦可再證得f_n是uniformly bdd.
pf：from above ： │f_n(x)-f(x)│< 1 , for all n>= N
by triangle ineqality , │f_n(x)│< │f(x)│ + 1 , for all n>=N
since f(x) is bdd. on S, then there exists M > 0
s.t. │f_n(x)│< M , for all n>=N
take M'=max{M,M_1,M_2,....M_(N-1)}
then │f_n(x)│< M' , for all n
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 114.25.179.183