Why are simple returns additive over securities? I've found a few sources for instance this one, that makes the claim that over a portfolio, simple returns are additive. I can see why it doesn't make a difference if we use net or gross returns as long as the weights add to one, but I can't see why we should get this additive behaviour for a quantity that has a division in it. Perhaps I am using a bad definition of weight?

If someone could share a derivation for a portfolio of two assets that would be sufficient.
We have a portfolio of two assets, with prices $P_1(0), P_2(0)$ at $t=1$. In the next period, their prices are $P_1(1), P_2(1)$. 

The portfolio market value at $t=0$ is $V(0) = W_1 P_1(0) + W_2 P_2(0)$. 

The portfolio weights $\omega_i$ (for $i =1,2$) are:
$$
\omega_i = \frac{W_i P_i(0)}{W_1 P_1(0) + W_2 P_2(0)}.
$$

(We can easily see that $\omega_1 + \omega_2 = 1$.)

The return for an asset $R_i$ is given by:
$$
R_i = \frac{P_i(1)}{P_i(0)} - 1.
$$

The market value of the portfolio $V(1)$ is equal to:
$$
V(1) = W_1 P_1(0)(1+R_1) + W_2 P_2(0) (1+R_2).
$$
Then,
$$
V(1) = (W_1 P_1(0) + W_2P_2(0))(1 + \frac{W_1P_1(0)R_1}{W_1 P_1(0) + W_2P_2(0)} +
\frac{W_2P_2(0)R_2}{W_1 P_1(0) + W_2P_2(0)}),
$$
$$
V(1) = V(0) (1 + \omega_1 R_1 + \omega_2 R_2).
$$
If we define the portfolio return $R$ as $\omega_1 R_1 + \omega_2 R_2$, then
$$
V(1) = V(0) (1 + R).
$$

This is easier to see if we normalise prices.