$f$ is integrable & $int ^a _b f= beta iff forall epsilon >0 exists mathbb{P}$ partition such as $U(f,P)-epsilon < beta < L(f,P)+ epsilon$

Under the Riemann sum definition, the reverse implication is proved as follows.

If for any $epsilon > 0$ there exists a partition $P_epsilon$ such that $beta - epsilon < L(f,P_epsilon) leqslant U(f,P_epsilon) < beta + epsilon$, then for any refinement $P supset P_epsilon$ and Riemann sum $S(f,P)$ we have

$$beta - epsilon < L(f,P_epsilon) leqslant L(f,P) leqslant S(f,P) leqslant U(f,P ) leqslant U(f,P_epsilon) < beta + epsilon$$

Hence, $|S(f,P) - beta | < epsilon$ for all $P supset P_epsilon$ and $int_a^b f = beta$.

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For the forward implication, if $int_a^b f = beta$, then there is a partition $P$ such that for any Riemann sum $S(f,P)$

$tag{*} beta - frac{epsilon}{2} < S(f,P) < beta + frac{epsilon}{2}$

For any partition subinterval $I_j =[x_{j-1},x_j] $, there exists $xi_j, eta_j in I_j$ such that

$$underbrace{sup_{x in I_j} f(x)}_{M_j} - frac{epsilon}{2(b-a)} < f(xi_j), quad f(eta_j)<underbrace{inf_{x in I_j} f(x)}_{m_j} + frac{epsilon}{2(b-a)},$$ and it follows that the Riemann sums with intermediate points ${xi_j}$ and ${eta_j}$ satisfy

$$tag{**}U(f,P) = sum_{j}M_j(x_j - x_{j-1}) < underbrace{sum_{j}f(xi_j)(x_j - x_{j-1})}_{ S(f,P,{xi_j})}+frac{epsilon}{2} , \ L(f,P) = sum_{j}m_j(x_j - x_{j-1}) > underbrace{sum_{j}f(eta_j)(x_j - x_{j-1})}_{ S(f,P,{eta_j})}-frac{epsilon}{2}$$

From (*) and (**) we get

$$beta - epsilon < S(f,P,{eta_j})- frac{epsilon}{2}< L(f,P) leqslant U(f,P) < S(f,P,{xi_j}) + frac{epsilon}{2} < beta + epsilon$$