Fractional parts of multiples of √2

Pigeonhole. Among $\{0\},\{1\sqrt2\},\ldots,\{N\sqrt2\}$ (which are $N+1$ points in $[0,1)$), two differ by at most $1/N$ in fractional part. Say $\{k\sqrt2\}$ and $\{\ell\sqrt2\}$ with $0<\delta=\{(\ell-k)\sqrt2\}<1/N$ (taking absolute difference).

Density. Since $\sqrt2$ is irrational, $\delta\neq0$, and multiples of $\delta$ mod $1$ form a $\delta$-dense set in $[0,1)$.

Conclusion. For any $N$, we get $\delta<1/N$, so the set is dense.