so for each $\phi_n$, instead set $x \in N$ to 0 (b/c the $\phi_n$ are the lower approximations, and $f$ assumed nonnegative) for $\psi_n$, set $x \in N$ to $M$, the upper bound for $f$. Solution to exercise 26 from chapter 7 from Gerald Folland's textbook, 'Real Analysis: Modern Techniques and Their Applications. if the function is a simple modification of $\sin, \cos, \exp$ (aka something you should know taylor series of), then expand out into taylor series.FTC: if $F(x) = \int f$ and $f’$ is integrable, then $F’(x) = \int f’$.this is because $|a_n - a| \geq \big| |a_n| - |a| \big|$ good trick to apply MCT: bump up everything by an integrable function apply MCT now i can cancel the integral i added, b/c it’s finite.Toc edu notes tangents log about folland ch 2
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