(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [tex]f[/tex] be a function that is integrable on [tex][a,b][/tex] and bounded by

[tex]0[/tex] [tex]\leq f(x)\leq M[/tex] for some [tex]M[/tex]. Prove that [tex]f^{2}[/tex] is also integrable on this interval.

2. Relevant equations

We've done many problems with step functions [tex]s(x)[/tex][tex]\leq f(x) \leq t(x)[/tex],

where [tex]s(x)[/tex] and [tex]t(x)[/tex] are step functions. Then we've defined a

function's lower integral as the supremum of the set of the values of all [tex]\int_a^b s(x)dx[/tex]

provided that [tex]s(x)[/tex] [tex]\leq f(x)[/tex] and the function's upper integral as the

infimum of the set of all [tex]\int_a^bt(x)dx[/tex] provided that

[tex]f(x)[/tex] [tex]\leq t(x)[/tex] on [tex][a,b][/tex].

3. The attempt at a solution

Attached is my work. It's a little sloppy, so I'll explain what I tried here:

I first tried using the weighted mean value theorem, but all that that does is show that [tex]f^{2}[/tex] is bounded.

I then drew a line and tried again. However, I've realized that my definition of the step functions is faulty because the function is not monotonic, so the second attempt is wrong. As a result, I've been unable to represent my step functions more explicitly using summation notation and expanding it, so I haven't been able to work with the equations. There must be something I'm missing.

If the function were continuous, then I'd know that [tex]f^{2}[/tex] is continuous and I'd be done, but unfortunately it is not.

The only thing I know is that [tex]\int_a^b f(x)dx[/tex] is bounded and exists, and so [tex]\int_a^b f(x)^{2}dx[/tex] is also bounded.

Again, if I can show that the lower integral and the upper integral are equal, then by our definition I will have proven that the function is integrable.

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# Homework Help: A Step Function Proof

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