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**manjohn12** But suppose that you did not have the notion of upper and lower Riemann sums?

**Proof**. Let $\displaystyle \epsilon >0 $. Choose $\displaystyle \delta < \frac{\epsilon}{2} $. Let $\displaystyle P = \{x_0, x_1, x_2, \ldots, x_n \} $ be a partition of $\displaystyle [a,b] $ with $\displaystyle ||P|| < \delta $. Choose $\displaystyle x_{1}^{*}, x_{2}^{*}, \ldots, x_{n}^{*} $ arbitrarily such that $\displaystyle x_{i-1} \leq x_{i}^{*} \leq x_{i} $. We know that $\displaystyle \mathcal{R}(f,P) = \sum_{i=1}^{n} f(x_{i}^{*})(x_{i}-x_{i-1}) $. Now $\displaystyle \mathcal{R}(f,P) = 0 $ if $\displaystyle x_{i}^{*} \in D $. If $\displaystyle x_{i}^{*} \notin D $, then it is a limit point of $\displaystyle D $. From here how would you conclude that $\displaystyle \int_{a}^{b} f = 0 $?