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討論串[分析] bounded vs. convergent subsequence
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假設 f_k monotone and uniformly bounded on [0,1]，則存在. 逐點收斂之子序列. 大致步驟如下：. 1 用對角線法，找出一個在有理數集上逐點收斂之子序列。仍將. 此子序列記為 f_k。. 2 不難看出，步驟一隻極限在有理數上遞增。將此極限延拓為一. [0,1
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According to your repost, codomain of each f_k is R (set of real numbers).. The answer is positive. You can prove directly as your repost, or by gener
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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
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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 t
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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,.... Q2:. f_k(x) uni
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