Re: [分析] bounded vs. convergent subsequence

[vata (va)]

※ 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,...




f_k (i) =  1    if k = 0 mod i

          1/2   if k = 1 mod i

          1/3   if k = 2 mod i

          1/4   if k = 3 mod i

          ......
          ......

          1/i   if k = i-1 mod i

maybe this f_k(i) can't find a f_{k_j} (i) converge for all i=1,2,...

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→ GSXSP       : f_k (3) k>3 is not defined?
→ vata        : f_k(i)=1/i if k = i-1 mod i  for i=1,2,3,4,....
→ GSXSP       : Then what is f_5(3)?
→ willydp     : 可以啊, k_j := j!的話, f_{k_j}(i) = 1 for j>i.