How do I prove or disprove the following statements: $$(a) ∀n ∈ \mathbb N, ∃k ∈ \mathbb N, ∀x ∈ \mathbb R , \lfloor nx \rfloor − n \lfloor x \rfloor ≤ k$$ and $$ (b)\exists k \in \mathbb N, \forall n \in \mathbb N, \forall x \in \mathbb R, \lfloor nx \rfloor − n \lfloor x \rfloor ≤ k $$
I've also been given the following three properties to use: $ (i) \forall x \in \mathbb R, \exists \epsilon \in R, 0 ≤ \epsilon < 1 ∧ x = \lfloor x \rfloor + \epsilon \\ (ii) \forall x \in \mathbb Z, \forall y \in \mathbb R, \lfloor x + y \rfloor = x + \lfloor y \rfloor \\ (iii) \forall x \in \mathbb Z, \lfloor x \rfloor = x $
I know that we must negate the statement$(a)$ first in order to pave the way for a disproof(if it false) by proving: $$ \exists n \in \mathbb N, \forall k \in \mathbb N, \exists x \in \mathbb R, \lfloor nx \rfloor - n \lfloor x \rfloor > k $$ Other than that I have no idea which property to use and in which order. The help would be really appreciated.