32 Proof of properties of exponential

(i) Note that for a matrix $X$ the entries $X_{jk}$ satisfy $|X_{jk}|\le \parallel X\parallel$. Also,

$$\parallel A^2\parallel \le {\parallel A\parallel }^2,\mathrm{\dots },\parallel A^m\parallel \le {\parallel A\parallel }^m,$$

so

$$p_m(A)=I+A+\frac{A^2}{2!}+\mathrm{\dots }+\frac{A^m}{m!}$$

satisfy

$$\parallel p_m(A)-p_k(A)\parallel \le \parallel \frac{A^{k+1}}{(k+1)!}+\mathrm{\dots }+\frac{A^m}{m!}\parallel \le \sum _{j=k+1}^m\frac{{\parallel A\parallel }^j}{j!}$$

where $e^{\parallel A\parallel }={\sum }_{j=0}^{\mathrm{\infty }}{\parallel A\parallel }^j/j!$ converges. Hence each entry of $p_m(A)$ converges as $m\to \mathrm{\infty }$, giving $\exp (A)$ as the limit.