Prove for a bounded function on $[a,b]$ that if $\phi=\Phi$, then for every $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that for all partitions $P$ where $||P||\lt\delta$, then $U(P)-L(P)\lt\epsilon$.
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Lesson Parent:
Problem:
Prove for a bounded function on $[a,b]$ that if $\phi=\Phi$, then for every $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that for all partitions $P$ where $||P\,||\lt\delta$, then $U(P)-L(P)\lt\epsilon$.
Answer:
It is true that if $\phi=\Phi$, then \[ \forall\epsilon>0(\exists\delta>0(\forall P(||P||<\delta\Rightarrow U(P)-L(P)<\epsilon))). \]
