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Suppose we are given the following: $$\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$$

How would we prove this? Does this follow from Cauchy Schwarz? Intuitively this is how I see it: In the LHS we could have a negative area that reduces the positive area. In the RHS the area can only increase because we take the absolute values of the functions first.

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The big idea here is this:

First: it is enough to show that

$$

\left\lvert\int_a^b f(x)\,dx\right\rvert\leq\int_a^b\lvert f(x)\rvert dx,

$$

since you can replace $f(x)$ by $f(x)\cdot g(x)$ to get the desired result.

Now, notice that

$$

-\lvert f(x)\rvert\leq f(x)\leq \lvert f(x)\rvert

$$

for all $x$; hence

$$

-\int_a^b\lvert f(x)\rvert\,dx\leq \int_a^b f(x)\,dx\leq\int_a^b\lvert f(x)\rvert\,dx.

$$

Can you finish it from here?

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