Consider the functions $f_n(z)=(1+\frac{z}{n})^n$ for each $n=1,2,3,...$ Use the fact that $f_n(x)$ $\rightarrow e^x $ for each $x \in \Bbb R $ to prove that $f_n(z) \rightarrow e^z$ normally in $\Bbb C$.
My work: Suppose $f_n(z)=(1+\frac{z}{n})^n$ for each $n=1,2,3,...$ Since $1+x \le e^x$ for $x \ge 0$, it follows that $|f_n(z)| \le e^{|z|}$, for all $z$. So, ${f_n}$ is a normal family. And $f_n(x)$ $\rightarrow e^x $ for each $x \ge 0$.
I am interested to perform following steps:
Since Uniformly Bounded Family is Locally Bounded, $\{f_n(z)\}$ is locally bounded on U. From Montel's Theorem, $\{f_n(z)\}$ is a normal family, From Vitali's Convergence Theorem, $\{f_n(z)\}$ converges uniformly on any compact subset of U. In particular, $\{f_n(z)\}$ converges uniformly on K. My intuition is to prove $f_n(z) \rightarrow e^z$ uniformly on compact sets. Your kind help will be appreciated. Thank you so much!