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?

It is important to be sure that things you are dealing with, like $\sqrt{2}$ or $e$, exist, otherwise correct manipulations with them will make no sense. Some examples:


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*Let $x$ be a solution to $x+1=x$. Then $1=0$.

*Perron paradox: Let $N$ be the largest positive integer. If $N>1$ then $N^2>N$, which contradicts the assumption. Hence, $N=1$.


When you are ordering dinner in a restaurant, you better prove first that the wallet exists in your pocket, otherwise you may easily get into trouble.
