Is this a mathematically valid method of creating a sequence of rationals that converges to an irrational, or is it a handwaving argument?
I know that I could create a sequence by actually giving a formula.
Let $x$ be an irrational. Choose rationals between $x-1/n$ and $x+1/n$ for all $n$.
I find it a bit suspicious because I'm not specifying clearly what rational I'll be choosing. I can't say choose the smallest/largest.