Fair point. 
I guess it depends on the audience. Like I think my footnote proof that every circle contains a point with rational coordinates inside of it is complete overkill for anyone who’s comfy with the idea that the rationals are dense in the reals.
It occurs to me that that fact is the one that wants the footnote! Here’s what it means to say that the rationals (ie, fractions) are dense in the reals (ie, decimals numbers): If you pick any two real numbers, a and b, and they’re not the same, then no matter how ridiculously close together they are, like a googolplexth apart or something, there’s a rational number strictly in between them.
One way to get an intuition for why that’s true is to start with the fairly obvious fact that if you can write a number as a terminating decimal (the digits stop at some point) then it’s rational. Like 0.1234567 is the fraction 12345467/100000000. (Repeating decimals like .3333… are also rational but that’s not relevant to this proof.) So with your 2 real numbers ridiculously close together, they might match for the first gazillionth decimal places but eventually they have to stop matching, and that means you can just go out that far and pick something in between them and then let that decimal terminate and, voila, that’s a rational number!
The fact that the rationals are dense in the reals and yet are countable whereas the reals are uncountable is pretty mind-blowing. 
Cantor