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jdcasey9
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Homework Statement
If f,g [tex]\in[/tex] R[tex]\alpha[/tex][a,b] with f[tex]\leq[/tex]g, show that [tex]\int[/tex]abfd[tex]\alpha[/tex] [tex]\leq[/tex] [tex]\int[/tex]abgd[tex]\alpha[/tex]
Homework Equations
Let [tex]\alpha[/tex]: [a,b] [tex]\rightarrow[/tex] R(Real numbers) be increasing. A bounded function f: [a,b] [tex]\rightarrow[/tex] R(real numbers) is in R[tex]\alpha[/tex][a,b] if and only if, given [tex]\epsilon[/tex][tex]\succ[/tex] 0, there exists a partition P of [a,b] such that U(f,P) - L(f,P) [tex]\prec\epsilon[/tex].
m=min{m1,..., mn}
M=max...
mi=inf{f(x) : xi-1[tex]\leq[/tex]x[tex]\leq[/tex]xi}
Mi=sup{f(x) : xi-1[tex]\leq[/tex]x[tex]\leq[/tex]xi}
The Attempt at a Solution
This problem doesn't seem to be difficult, but I'm running into problems trying to compare f and g:
For f and g: m[tex]\leq[/tex] mi [tex]\leq[/tex] L(f,P) [tex]\leq[/tex][tex]\int[/tex]b_a [tex]\leq[/tex][tex]\int[/tex]_ba [tex]\leq[/tex] U(f,P) [tex]\leq[/tex] Mi [tex]\leq[/tex] M
So we can say such things as:
mf [tex]\leq[/tex] mg for each piece, but can we say that:
mf [tex]\leq[/tex] mg [tex]\leq[/tex] mfi [tex]\leq[/tex] mgi ... etc.
If we can then this is easy to prove. If not, then how do we relate different parts of f and g?
Thanks.
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