If $x$ is a positive real number, and $n$ is a positive integer, prove that \[[ nx] > \frac{[ x]}1 + \frac{[ 2x]}2 +\frac{[ 3x]}3 + \cdots + \frac{[ nx]}n,\] where $[t]$ denotes the greatest integer less than or equal to $t$. For example, $[ \pi] = 3$ and $\left[\sqrt2\right] = 1$.