33 Proof of properties of exponential, continued

(ii) We write

$$p_{2m}(zA)p_{2m}(wA)=\sum _{j=0}^{2m}\frac{z^j{A}^j}{j!}\sum _{k=0}^{2m}\frac{w^k{A}^k}{k!}$$

and compare with

$$p_m((z+w)A)=\sum _{r=0}^m\frac{{(z+w)}^r{A}^r}{r!}=\sum _{r=0}^m\sum _{s=0}^r\frac{z^s{w}^{r-s}{A}^r}{s!(r-s)!}$$

where $p_{2m}(zA)\to \exp (zA)$, $p_{2m}(wA)\to \exp (wA)$, and $p_m((z+w)A)\to \exp ((z+w)A)$, so $\exp ((z+w)A)=\exp (zA)\exp (wA).$