convergence

**1234567** · October 26th 2010, 11:40 AM

Can someone show me the proof for the following:

**emakarov** · October 26th 2010, 02:47 PM

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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