Prove: Every collection of r subsets each with atleast $|X|(1-\frac{1}{r})$ points of X has a non empty intersection.

Question

I am unable to show this "trivial" fact:

Let $X$ be a finite set. Then every family of subsets of $X$ of size $r \ge 2$, where each subset contains atleast $|X|(1-\frac{1}{r})$ points of $X$ has a non-empty intersection in X.

My only idea was to use the inclusion-exclusion principle. So if $A_i$ are the r sets, then: \begin{align*} |\cup_i A_i| \le \sum_i |A_i| + (-1)^{r+1} |\cap_i A_i| \end{align*} which gives nothing.