Consider the function $e^x$ on the reals. I want to show that $e^x$ is continuous at the any point $t \in \mathbb{R}$. Is the following argument valid?
(1) Let $\{t_n\}$ be an arbitrary sequence of reals s.t. $t_n \to t$ and $t_n \ne t$.
(2) Then $\lim\limits_{t_n \to t}$ $e^{t_n} = \lim\limits_{t_n \to t}1 + \frac{{t_n}^2}{2} + \frac{{t_n}^3}{6} + \ldots = 1 + \frac{{t}^2}{2} + \frac{{t}^3}{6} + \ldots = e^t$
So that since $\{t_n\}$ is arbitrary it follows $e^x$ is continuous at $t$.