definition of integration using equally spaced intervals if $f$ is Riemann integrable then $\exists$ sequence $D_{n}$ of dissections such that $S_{D_{n}} - s_{D_{n}}$ tends to $0$ 
is the above equivalent to saying:
consider ${D_{n}} = a + \frac{k(b-a)}{n} $
interval is $[a,b]$ $0 \leq k \leq n$ 
$f$ is integrable if there exists a N such that if n $\geq$ N then $S_{D_{n}} - s_{D_{n}} \leq \epsilon$
in words: a function is integrable if there exists a dissection with equally spaced points such that the upper and lower sum is less than a given $\epsilon$?
I realise you can always add points to a dissection but can you move points such that the points are equally spaced over the interval?
 A: Yes.
In the Riemann-Darboux context, a bounded function is integrable if it satisfies the Riemann criterion. There exists some partition (dissection) $P_\epsilon$ given any $\epsilon > 0$ such that the difference between the upper and lower sums is less than $\epsilon:$ 
$$U(P_\epsilon,f) - L(P_\epsilon,f) < \epsilon.$$
How the partition points are distributed is not relevant.
Thus, if the sequence as you define it, with or without uniformly spaced points, tends to $0$, then for any $\epsilon > 0$ there exists $N \in \mathbb{N}$ such that
$$S_{D_N} - s_{D_N} = U(D_N,f) - L(D_N,f) < \epsilon.$$
Then it can be shown that the lower and upper Darboux integrals are  equal
$$ I = \sup_{P} L(P,f) = \inf_{P} U(P,f).$$
This is equivalent to there exists a number $I$ such that for any $\epsilon >0$ there exists a partition $P_\epsilon$ such that for any refinement $P$ ($P_\epsilon \subset P$) we have for any Riemann sum (regardless of the choice of intermediate points)
$$|I - S(P,f)| < \epsilon.$$
This is equivalent to there exists a number $I$ such that for any $\epsilon >0$ there exists $\delta > 0$ such that for any partition $P =(x_0,x_1, \ldots, x_n)$ with mesh less than $\delta$ ( $\max_{1 \leqslant i \leqslant n}(x_i - x_{i-1})< \delta )$we have for any Riemann sum (regardless of the choice of intermediate points) 
$$|I - S(P,f)| < \epsilon.$$
