Let $n$ be an integer (n>1). Show that there exists a proper subset $A$ of $\{1,2,\cdots, n\}$ such that the following holds:
the numbers of elements of $A$ is no more than $2[\sqrt n]$+1. ([x] means the greatest integer which is no more than x).
$\{\mid x-y\mid: x,y\in A, x\neq y\}=\{1,2,\cdots, n-1\}$