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