※ 引述《GSXSP (Gloria)》之銘言：
: Q1:
: f_k (i) uniformly bounded for all i = 1, 2 ,...
: Can we find a subsequence k_j such that
: f_{k_j} (i) converge for all i=1,2,...
By weinerstress,
for f_k (1) we have subsequence 1,k
k_k (i) we have subsequence i,k
Take subsequence k,k then it converges for all i=1,2,...
is this correct?
: Q2:
: f_k(x) uniformly bounded for all x \in [0,1]
: Can we find a subsequence k_j such that
: f_{k_j} (x) converge for all x \in [0,1]
Anyone can provide counter-example?
What if further f_k(x) are increasing and cts functions?
(
https://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem
requires equi-cts and getting uniformly convergence
but I have only increasing and cts and only need pointwise convergence
)
----------
To be specific, I think the following might be true but I cannot find a
proper theorem to support it, neither a counter-example
Statement:
For a sequence of increasing and uniformly bounded functions on compact set,
we can always find a convergent subsequence in the sense of sup-norm
can anyone help?
--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 108.193.239.125
※ 文章網址: https://www.ptt.cc/bbs/Math/M.1468085238.A.F23.html
※ 編輯: GSXSP (66.75.244.69), 07/10/2016 09:23:08