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Math Help - convergence

  1. #1
    Junior Member
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    convergence

    Can someone show me the proof for the following:

    convergence-untitled.jpg
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  2. #2
    MHF Contributor
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    Is B[0,1] the set of bounded functions on [0,1], and is d_\infty(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|? Then show that for each fixed x, |f_n(x)-x|=|(f_{n-1}(x)+x)/2-x|\le1/2|f_{n-1}(x)-x|. This would imply that d_\infty(f_n,I)=\sup_{x\in[0,1]}|f_n(x)-x|\le1/2\sup_{x\in[0,1]}|f_{n-1}(x)-x|=1/2d_\infty(f_{n-1},I).
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