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1 1 $begingroup$ There are many ways to define the Riemann Integral. I am using this one, where I denote $sigma(f,P^{*})$ the Riemann Sum relative to a tagged partition $P^{*}$ : $textbf{Definition}$ We say that a function $f:[a,b] to mathbb{R}$ is Riemann-Integrable if exist the limit: $I=lim_{||P|| to 0} sigma(f,P^{*})$ and then we write $I=int_{a}^{b}f(x)dx$ . The limit exist in the sense that given $epsilon > 0$ , there is $delta > 0$ such that for any partition $P$ of $[a,b]$ with $||P|| < delta$ and for any tagged partition $P^{*}$ , we have: $$|sigma(f,P^{*}) - I| < epsilon$$ By definition, if $f$ is integrable in $[a.b]$ , then given $epsilon < 0 $ exists two tagged partition $P^{*}$ and $P^{**}$ such that: $$|sigma(f,P^{*}) - I| < epsilon / 2$$ $$|sigma(f,P^{**