How to show that the collection of Borel-measurable functions from $\mathbb{R}$ to $\mathbb{R}$ is the smallest one with these properties? 
Consider collections $ \mathcal{V} $ of functions from $ \mathbb{R} $ to $\mathbb{R} $ satisfying the following conditions:
(a) $ \mathcal{V} $ is a vector space
(b) $ \mathcal{V} $ contains the continuous functions
(c) If $ (f_{n})_{n} $ is an increasing sequence of nonnegative functions in $ \mathcal{V} $ and if $ \lim_{n \rightarrow \infty} f_{n}(t) $ exists and is finite for all $ t \in \mathbb{R} $, then $ \lim_{n\rightarrow \infty} f_{n}(t) \in \mathcal{V} $ .
Show that the collection $ \mathcal{V}_{0} $ consisting of the Borel-measurable functions is the smallest such collection of functions. (Hint: define $\mathcal{A}= \{ A \subseteq \mathbb{R}: \chi_{a} \in \mathcal{V} \}$. Show that $ \mathcal{A} $ contains the interval $ (-\infty,a) $, and then contains the Borel sets).

$\chi_{A} :$ characteristic function of $A$.
$\chi_{A}(x)=1$ if $x \in A $ , $\chi_{A}(x)=0$ if $x \notin A, $
I need to find or prove the existence of a sequence of positive continuous functions, increasing to converge to $ \chi_ {A} $, then I could use the basic theorem of measure theory to put all the functions borel measurable in the set. Any help is appreciated, I try with step functions but dont result.
 A: It is not true that every $\chi_A$, where $A$ is Borel, can be written as a limit of continuous functions.  (Not even $\chi_\mathbb{Q}$ can be written this way.)
Consider the set $\mathcal{A}$ defined in the hint.  As Norbert's answer suggests, you can show $(-\infty, a) \in \mathcal{A}$ for each $a \in \mathbb{R}$.  Now use  Dynkin's $\pi$-$\lambda$ theorem.  Show that $\mathcal{A}$ is a Dynkin system (or $\lambda$-system).  The collection $\mathcal{P} = \{(-\infty, a) : a \in \mathbb{R}\}$ is a $\pi$-system contained in $\mathcal{A}$, so Dynkin's theorem says $\mathcal{A}$ contains $\sigma(\mathcal{P})$, which is exactly the Borel $\sigma$-algebra.
Once this is done, you have $\chi_A \in \mathcal{V}$ for every Borel set $A$.  Then $\mathcal{V}$ contains all the simple functions (finite linear combinations of characteristic functions), and any Borel measurable function is a pointwise limit of simple functions.
What you are proving here is essentially a special case of the functional monotone class theorem.
A: Using the hint it is enough to prove that $\chi_{(-\infty,a)}$ can be pointwise limit of continuous increasing functions. Indeed, consider functions
$$
f_n(x)=
\begin{cases}
1 & \text{ if }\quad x<a-1/n\\
n(a-x) & \text{ if }\quad a-1/n\leq x<a\\
0 & \text{ if }\quad x\geq a
\end{cases}
$$
