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## Homework Statement

## Homework Equations

## The Attempt at a Solution

Right now, I'm still trying to understand why the hint is true. This is what I've got so far...

Let ||f||

_{∞}= sup{|f(x)|: x E [a,b]}

[tex]M_i(f,P)[/tex] = sup{f(x): [tex]x_{i - 1}[/tex] ≤ x ≤ [tex]x_i[/tex]}

[tex]m_i(f,P)[/tex] = inf{f(x): [tex]x_{i - 1}[/tex] ≤ x ≤ [tex]x_i[/tex]} where P is a partition of [a,b]

Let x,t E [[tex]x_{i - 1}, x_i[/tex]]

Then |f(x)g(x)-f(t)g(t)| ≤ |f(x)| |g(x)-g(t)| + |f(x)-f(t)| |g(t)|

≤ ||f||

_{∞}[[tex]M_i(g,P) - m_i(g,P)[/tex]] + [[tex]M_i(f,P) - m_i(f,P)[/tex]] ||g||

_{∞}

How can we finish proving the hint from here? I have no idea how to get M

_{i}(fg, P) - m

_{i}(fg, P) on the LHS of the inequality...

I hope somebody can help me!

Any help is much appreciated!