I find the exponential function extremely hard to grasp from a rigorous point of view. For example, I want to show that $0 \le e^x \le 1$ for $x \in \mathbb{R}_{\le 0}$.
First, if $x = 0$, then $e^x = e^0 = \Sigma_{n=1}^\infty \frac{0^n}{n!}= 0 + \frac{0^2}{2} + \ldots = 0$.
Otherwise if $x < 0$ then we have
$e^x = \Sigma_{n=1}^\infty \frac{x^n}{n!}= x + \frac{x^2}{2} + \frac{x^3}{6} \ldots$
Now clearly the odd terms in this sequence are negative while the even terms of this sequence are positive. But it's not clear to me why this sequence will always be bounded by $1$. For example, if we took out the negative terms the sequence could certainly be larger than $1$ (so we can't bound $e^x$ by its positive terms).
What then are then some useful bounds with the exponential function to solve problems like this and related problems? What if we fixed $t > 0$ and considered for example $e^{tx}$?