What is the purpose of showing some numbers exist? For example in my Analysis class the professor showed $\sqrt{2}$ exists using Archimedean properties of $\mathbb{R}$ and we showed $e$ exists. I want to know why it's important to show their existence?

You have to start with definitions of the real numbers and real number multiplication. Then you must use set theory and the rules of logic to construct a set $x$ and prove (a) that $x$ is a real number by your definition, and (b) that $x\cdot x = 2$ by your definition of multiplication. Only then are you entitled to say that there exists a real number $x$ such that $x\cdot x = 2$.
