Proof of Forward Euler for discretizing a transfer function
Another way to see how the forward Euler method approximates a continuous-time system is by considering the "ideal" mapping of the $s$-plane to the $z$-plane (why?):

$$z=e^{sT}\tag{1}$$

For frequencies that are much smaller than the sampling frequency (i.e., $|s|T\ll 1$) we can approximate $e^{sT}$ by its first order Taylor series:

$$z\approx 1+sT\tag{2}$$

which is exactly the forward Euler mapping, resulting in

$$s=\frac{z-1}{T}\tag{3}$$