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Prove for a bounded function on $[a,b]$ that there exits a unique number $I$ such that for all partitions $P$ , $L(P)\leq I\leq U(P)$ if and only if $\phi=\Phi$ .
Prove for a bounded function on $[a,b]$ that if $\phi=\Phi$ , then there exits a unique number $I$ such that for all partitions $P$ , $L(P)\leq I\leq U(P)$ .
Prove for a bounded function on $[a,b]$ that if there exists a unique number $I$ such that for all partitions $P$ , $L(P)\leq I\leq U(P)$ , then $\phi=\Phi$ .
Prove for a bounded function on $[a,b]$ that $\phi=\Phi$ if and only if for every $\epsilon\gt 0$ there exists a $\delta\gt 0$ such that for all partitions $P$ where $||P\,||\lt\delta$ , then $U(P)-L(P)\lt\epsilon$ .
Prove, for a bounded function on $[a,b]$ , that if $\forall\epsilon\gt 0(\exists\delta\gt 0(\forall P(||P\,||\lt\delta\Rightarrow U(P)-L(P)\lt\epsilon)))$ is true, then $\phi=\Phi$ .
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 $U(P)-L(P)\lt\epsilon$ .
Prove that every upper sum has a greatest lower bound.
Prove that the set of all lower sums has a least upper bound.
Prove that if $f(x)$ is bounded on $[a,b]$ , then $\lim_{||P\,||\to 0}\,L(P)$ is the least upper bound of all lower sums.
Prove that if $f(x)$ is bounded on $[a,b]$ , then $\lim_{||P\,||\to 0}\,U(P)$ is the greatest lower bound of all upper sums.
Prove that $\phi\leq\Phi$ where $\phi$ is the least upper bound of all lower sums and $\Phi$ is the greatest lower bound of all upper sums.

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GAIN AN ADVANTAGE

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