NL-NWO 1967 Problem 5

Consider rows of the form: $[x], [2x], [3x], ...$ Proof that, if $N \in N$ does not occur in the sequence $([n x])$ , then there is an $n \in N$ with $n - 1 < \frac{N}{x}< n -\frac{1}{x}$ Prove that, for $x, y \notin Q$ : $\frac{1}{x}+\frac{1}{y} = 1$ , then each $N \in N$ term is either of $([nx])$ or of $([ny])$ .