$f$ is integrable & $\int ^a _b f= \beta \iff \forall \epsilon >0 \exists \mathbb{P}$ partition such as $U(f,P)-\epsilon < \beta < L(f,P)+ \epsilon$ Let $f$ be bounded in $[a,b]$. Prove $f$ is integrable and
$\int ^a _b f= \beta \iff \forall \epsilon >0 \ \exists \ \mathbb{P}$ partition such as $U(f,P)-\epsilon < \beta < L(f,P)+ \epsilon$
I already saw that these are equivalent:
a. $f$ is integrable in $[a,b]$
b. $\forall \ \epsilon >0 \ \exists \ Q \ \in \mathbb{P} :$ $$\forall  \ P \in \mathbb{P} \ \ Q \subseteq P \Rightarrow U(f,P) -L (f,P) < \epsilon$$
c. $\forall \ \epsilon >0 \ \exists \ Q \in \mathbb{P}:$
$$U(f,Q)-L(f,Q)<\epsilon$$
 A: 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 $\int_a^b f = \beta$.

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