Question: Let f be a nonnegative bounded function on [a, b] with \(0\le f\le M\). Let

for n = 1, 2, … and \(k=0,1,2,...,{{2}^{n}}\) and set \({{\varphi }_{n}}=\sum\nolimits_{k=0}^{{{2}^{n}}}{\left( kM/{{2}^{n}} \right){{\chi }_{{{E}_{n,k}}}}}\). Prove that \(0\le {{\varphi }_{n}}\le {{\varphi }_{n+1}}\le f\) and that \(0\le f-{{\varphi }_{n}}\le {{2}^{-n}}M\) for each n . Thus \(\left( {{\varphi }_{n}} \right)\) converges uniformly to f on [a, b].

(Note that in this Exercise 1 there is \({{\chi }_{{{E}_{n,k}}}}\). Given a subset A of some "universal" set S, we define \({{\chi }_{A}}\) : S R , the characteristic function of A, by \({{\chi }_{A}}\) ( x ) = 1 if x is in A, and \({{\chi }_{A}}\) ( x ) = 0 if x is not in A.)

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