The following lemma and proof is from Real Analysis with Real Applications by Davidson and Donsig:
Lemma 6.3.2. If $R$ is a refinement of $P$, then \begin{align*} L(f,P) \le L(f,R) \le U(f,R) \le U(f,P) \end{align*}
Let $f(x)$ be a bounded function on $[a,b]$. The following are equivalent:
$(1)$ f is Riemann integrable.
$(2)$ For each $\epsilon >0$, there is a partition $P$ so that $U(f,P)-L(f,P)<\epsilon$
Proof $(1) \implies (2)$: If $f$ is Riemann integrable, let $L=L(f)=U(f)$. Let $\epsilon >0$. We can find two partitions $P_{1}$ and $P_{2}$ so that $U(f,P_{1}) < L + \dfrac{\epsilon}{2}$ and $L(f,P_{2}) > L-\dfrac{\epsilon}{2}$. Let $P$ be their common refinement, $P_{1} \cup P_{2}$. By Lemma 6.3.2., \begin{align*} L-\dfrac{\epsilon}{2}<L(f,P)\le U(f,P) < L + \dfrac{\epsilon}{2} \end{align*} and so $U(f,P)-L(f,P) \le \epsilon$, proving $(2)$.
How do I show that $U(f,P)-L(f,P) \le \epsilon$?