Let X be any real number. Prove that among the numbers X, 2X, 3X,.....(n-1)X there exists one such number that differs from an integer by at most 1/n.

Question

I have been trying to do this problem using pigeon hole principle but still now unable to find any elegant and easy solution to this problem. Any good or simple solution is appreciated.

Answer 1

If none of the fractional parts of $\{x\}, \{2x\}, \ldots,\{(n-1)x\}$ falls into $[0,\frac1n)$ or $[1-\frac1n,1)$, then thy fall into $n-2$ bins $[\frac1n,\frac2n)$ up to $[\frac{n-2}n,\frac{n-1}n)$ and hence we find two in one bin, i.e., $\{ rx\},\{ sx\}\in[\frac kn,\frac{k+1}n)$ with $0<r<s<n$. If $\{ rx\}\le \{ sx\}$, this means $\{(s-r)x\}\in[0,\frac1n)$, whereas if $\{ rx\}> \{ sx\}$, we have $\{(s-r)x\}\in[\frac1n,1)$.