Increasing sequence of simple functions to a bounded measurable function f

Assume $|f(x)|\leq M$, where $M\in\mathbb{N}$. Consider $\displaystyle g_k=\sum_{i=-M2^k+1}^{M2^k}\frac{i-1}{2^k}\chi_{\{\frac{i-1}{2^k}
Thus $\displaystyle g_{k+1}=\sum_{i=-M2^{k+1}+1}^{M2^{k+1}}\frac{i-1}{2^{k+1}}\chi_{\{\frac{i-1}{2^{k+1}}

For $x\in\{\frac{i-1}{2^k}, $g_k(x)=\frac{i-1}{2^k}$.

The above set is equal to $\{\frac{2i-2}{2^{k+1}}, so $g_{k+1}(x)\geq\frac{2i-2}{2^{k+1}}=g_k(x)$.

$|g_k(x)-f(x)|\leq\frac{1}{2^k}\to 0$ as $k\to\infty$. Hence $g_k$ converges to $f$ everywhere.

Author: mathtuition88

https://mathtuition88.com/

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