I have some non-integer real number (lets call it $b$), then there is some integer (lets call it $i$) that has the the property $i > b-1$ and $i < b$.
I believe this is a floor function (hopefully my terminology isn't wrong). I was curious on if my thought process on proving it made sense.
Essentially I'd set $b$ to some non-integer number like $9.85$ and then prove it by saying that if $i = 9$ then within the properties given there would have to be an integer between b and $b-1$. In this case it's $9$. I believe this is called a proof by example.