What does it mean when someone says that the exponential series converges uniformly on every bounded subset of the complex plane. I know the definition of convergence, uniform convergence of sequence of functions but I am a bit confused as to what does mean for an exponential series to be uniformly convergent
I am reading the Rudin's Real and complex analysis and this was the 2 line in the prologue of the book Thanks
Choose an arbitrary bounded set $A \subset \mathbb{C}$ and keep it fixed. Consider the sequence of partial sums $f_n$ defined by $f_n(z) = \sum_{j = 0}^n \frac{z^j}{j!}$ for all complex numbers $z$. This is a sequence of functions that converges to $f(z) = e^z$ for all $z \in \mathbb{C}$. The statement then is that this convergence is uniform on the set $A$ which had been chosen arbitrarily, that is $$ \forall \varepsilon > 0 \, \exists N \ge 0 \, \ni \forall z \in A \, \forall n \ge N \, |f_n(z) - f(z)| < \varepsilon \, . $$
Note that $N$ depends on $\varepsilon$ and also on $A$.