Definition 1:
$f$ is RI on $[a,b]$, $\exists L \in R$ ($L = \int_{a}^{b} f(x) dx$)
s.t. $\forall \epsilon > 0, \exists \delta > 0, \forall$ marked partition $P, \ if \ w(P) < \delta, \ then \ |S(f,P)- L| \leq \epsilon$
with $x_i \leq x_{i}^{*} \leq x_{i + 1}, \ i = 0,\dots, n - 1$
$P = \{a,x_1,\dots,x_{n-1},b\}$
Definition 2:
$f: [a,b] \to R$ is RI iff
$\forall \epsilon > 0, \exists \delta > 0 \ s.t. \ \forall$ marked partition
$P_1 = \{\dots x_1 \dots \}$ where $x_i \leq x_{i}^{*} \leq x_{i + 1}$
$P_2 = \{\dots y_i \dots \}$ where $y_i \leq y_{i}^{*} \leq y_{i + 1}$
if $w(P_1), w(P_2) < \delta$ then, $|S(f,P_1) - S(f,P_2))| \leq \epsilon$
Show that Definition 1 implies Definition 2.
I'm not quite sure how to start this. I have the idea of why it is true (the first one is kind of like the definition of continuity and the second is like the Cauchy definition). But I'm not sure how to put my thoughts into a proof for this. Any help is appreciated!
Edit: I actually did the proof the other way around. I know how to do it from Definition 2 to Definition 1. You can add and subtract L, and then use the triangle inequality. Unfortunately, the problem is asking for Definition 1 to imply Definition 2. So I need to do it that way.