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## Homework Statement

Suppose that [tex]f(x)[/tex] is a bounded function on [tex][a,b][/tex]. If [tex]M, M'[/tex] denote the least upper bounds and [tex]m, m'[/tex] denote the greatest lower bounds of [tex]f, |f|[/tex] respectively, prove that [tex]M'-m'\leq M-m[/tex].

**2. The attempt at a solution**

(these are the only things I have; most are properties found online)

For this, we assume that [tex]f[/tex] is integrable.

If [tex]f[/tex] is a bounded and integrable function on [tex][a,b][/tex], and if [tex]M[/tex] and [tex]m[/tex] are the least upper and greatest lower bounds of [tex]f[/tex] over [tex][a,b][/tex], then

[tex]m(b-a)\leq \int_a^b f(x)dx\leq M(b-a)[/tex] if [tex]a\leq b[/tex], and

[tex]m(b-a)\geq \int_a^b f(x)dx\geq M(b-a)[/tex] if [tex]b\leq a[/tex].

Also, since [tex]f[/tex] is both a bounded and integrable function on [tex][a,b][/tex], then [tex]|f|[/tex] is also bounded and integrable over [tex][a,b][/tex].

I haven't been able to determine how to obtain the least upper and greatest lower bounds of [tex]|f|[/tex], due to how complicated the very idea is.