Prove that $f$ is Riemann integrable iff $\forall ε > 0$ there exists a partition $P$ of $[a, b]$ s.t $U(f, P) − L(f, P) < ε$. Question:
Let $f : [a, b] → R$ be bounded. Prove that $f$ is Riemann integrable on $[a, b]$ if
and only if for each $ε > 0$ there exists a partition $P$ of $[a, b]$ such that $U(f, P) − L(f, P) < ε$.
Proof:
(⇒) Let $f$ be Riemann integrable on $[a, b]$ and let $ε > 0$. Then $\frac{ε}{2} > 0$ so
there exist partitions $P_1$ and $P_2$ of $[a, b]$ such that 
$L(f, P_1) > \mathcal{L} (f) − \frac{ε}{2}$ and $U(f, P_2) < \mathcal{U}(f) + \frac{ε}{2}. $
Let $P = P_1 ∪ P_2$. Then
$$U(f, P) − L(f, P) ≤ U(f, P_2) − L(f, P_1) < (\mathcal{U}(f) + \frac{ε}{2}) − (\mathcal{L}(f) − \frac{ε}{2}) = ε,$$
since $\mathcal{L}(f) = \mathcal{U}(f)$.
(⇐) Conversely, suppose that $f$ is not Riemann integrable on $[a, b]$. Let $ε = \mathcal{U}(f) − \mathcal{L}(f)$. Then $ε > 0$ but, for any partition $P$ of $[a, b]$,
$U(f, P) − L(f, P) ≥ \mathcal{U}(f) − \mathcal{L}(f) = ε$.
How can I conclude the second part? I don't really understand how the last step implies the statement.
 A: The definition of upper and lower integrals is $\mathcal{U}(f) = \inf_{P'} U(f,P')$ and $\mathcal{L}(f) = \sup_{P'} L(f,P')$ where the infimum and supremum are taken over all partitions $P'$ of $[a,b]$.
In other words, the upper integral $\mathcal{U}(f)$ is the greatest lower bound of all upper sums and for any particular partition $P$ we have $U(f,P) \geqslant  \mathcal{U}(f)$. Similarly we have $L(f,P) \leqslant \mathcal{L}(f)$.
Thus, 
$$\tag{*}U(f,P) - L(f,P) \geqslant \mathcal{U}(f) - \mathcal{L}(f)$$
In proving the reverse implication you first assume that $f$ is not Riemann integrable. This implies that $\mathcal{U}(f) > \mathcal{L}(f)$ and so $\alpha := \mathcal{U}(f) - \mathcal{L}(f) > 0$.  
Take $\epsilon = \alpha/2$. By (*)  it follows that for any partition $P$ we have
$$U(f,P) - L(f,P) \geqslant \mathcal{U}(f) - \mathcal{L}(f) = \alpha > \frac{\alpha}{2} = \epsilon$$
This contradicts the hypothesis that for any $\epsilon$ there is a partition $P$ such that $U(f,P) - L(f,P) < \epsilon$. Therefore, $f$ must be Riemann integrable.
