The limit $\lim_{|P| \to 0} U(f,P) $ exists I want to prove this :
Let $f:[a,b] \to \mathbf{R}$ a function bounded .For a particular partition $P$ of $[a,b]$ , $U(f,P) $ denotes the corresponding superior sum , then 
$\lim_{|P| \to 0}U(f,P)$ exist.
I know that if $P \subset P'$ thus $U(f,P')\le U(f,P)$ and it's plausible that $$U(f,P) \to \overline{\int_{a}^{b}}f\,dx\text{ as } |P|\to 0$$ 
My question is : How can I determine  $\delta$ for any $\epsilon$  such that  if $|P|<\delta$ then 
$$U(f,P) - \overline{\int_{a}^{b}}f\,dx < \epsilon \text{ ?}$$ 
 A: I think that you will be able to get an answer if you go through the proof of equivalence of two possible definitions of Riemann integral at Wikipedia. (I'll add a link to the current revision of the Wikipedia article, in case it changes in the future.)
One definition says that $s$ is the Riemann integral of $f(x)$ over the interval $[a,b]$ if, for each $\varepsilon>0$ there exists a $\delta>0$ such that for each partition with the mesh $m(P)<\delta$ the Riemann sum will be $\varepsilon$-close to $s$.
The other definition requires that for each $\varepsilon>0$ there is a partition $P$ such that for each refinement $P'\prec P$ the corresponding Riemann sum is $\varepsilon$-close to $s$.
Formulations for (upper and lower) Darboux integral would be analogous.
Both definitions can be considered as a special case of nets, but different preorder on the set of partitions is taken in each case. 
Since you have asked specifically about the choice of $\delta$, you might notice that in the proof in the Wikipedia article they choose
$$\delta < \min \left \{\frac{\varepsilon}{2r(m-1)}, (y_1 - y_0), (y_2 - y_1), \cdots, (y_m - y_{m-1}) \right \}$$
where $y_0,\dots,y_m$ is a partition for which the sum is $\varepsilon$-close to the integral and $r = \sup_{x \in [a, b]} |f(x)|$.
