Not use Lemma to prove the Borel field B(R)=σ({(a,b): -∞Prove 

Here, I have found some relative info:
Definition of Borel field:

Lemma:

But if do NOT use Lemma, how can I prove the above Borel field B(R)=σ({(a,b):-∞< a< b < ∞})?
Here are the links of someone else's working on the relative question: 
Prove the Borel field
Borel sigma field
 A: To prove that $D=E$ you prove that $D$ is contained in $E$ and $E$ is contained in $D$. In this case, you want to show that $\sigma(\mathcal A)=\sigma(\mathcal C)$ where $\mathcal A$ is the collection of (bounded) open intervals, and $\mathcal C$ is the collection of half-closed intervals $(-\infty,x]$ with $x\in\mathbb R$. 
In general, note first that if $\mathcal D\subseteq\sigma(\mathcal E)$, then also $\sigma(\mathcal D)\subseteq\sigma(\mathcal E)$. For the case at hand, this means that what you need to show is that $\mathcal A\subset\sigma(\mathcal C)$ and $\mathcal C\subset\sigma(\mathcal A)$. That is: Show that any open interval $(a,b)$ is in the sigma algebra generated by the intervals $(-\infty,x]$, $x\in\mathbb R$ and, conversely, show that any such half-closed interval is in the sigma algebra generated by the open intervals.
For instance, $(-\infty,x]$ is the complement of $(x,\infty)$, which we can rewrite as the countable union $\bigcup_n(x,x+n)$. This shows that each $(-\infty,x]$ is in $\sigma(\mathcal A)$. The other containment is established similarly.
