Can you prove that for any real number there exists as unique integer such as,
thus, the function is well-defined.
Well... there certainly exists one such number, n(x); Or else, the naturals would be bounded in the reals, something the Archimedean Property denies.
So there is one, at least. The set {n(x): n(x)-1< x <n(x)} must have a least element, by the Well-Ordering Property of the naturals. This least element, is exactly [x].
Personally, I would not bother myself so much just take x, and kill its integer part!