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 $\phi=\Phi$.

Problem: 

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 $\phi=\Phi$.

Answer: 

When \[ \forall\epsilon\gt 0(\exists\delta\gt 0(\forall P(||P\,||\lt\delta\Rightarrow U(P)-L(P)\lt\epsilon))) \] is true, then $\phi=\Phi$.