[All Steps] Let f be a nonnegative bounded function on $[a, b]$ with 0 ≤q f ≤q M. Let E_n, k=(k M)/(2^n) ≤q f for each n=1,2, ..., and k=0,1, ...,

Question: Let $f$ be a nonnegative bounded function on $[a, b]$ with $0 \leq f \leq M$. Let

\[E_{n, k}=\left\{\frac{k M}{2^{n}} \leq f<\frac{(k+1) M}{2^{n}}\right\}\]

for each $n=1,2, \ldots$, and $k=0,1, \ldots, 2^{n}$, and set $\varphi_{n}=\sum_{k=0}^{2 *}\left(k M / 2^{n}\right) X_{E_{x, 4}} .$

Prove that $0 \leq \varphi_{n} \leq \varphi_{n+1} \leq f$ and that $0 \leq f-\varphi_{n} \leq 2^{-n} M$ for each $n$. Thus, $\left(\varphi_{n}\right)$ converges uniformly to $f$ on [a, b]. [Hint: Notice that $E_{n, k}=E_{n+1,2 k} U$ $\left.E_{n+1,2 k+1} .\right]$

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[All Steps] Let f be a nonnegative bounded function on $[a, b]$ with 0 ≤q f ≤q M. Let E_n, k=(k M)/(2^n) ≤q f for each n=1,2, ..., and k=0,1, ...,

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