Simple Proof of Obvious Fact. Have you any idea of how to proof the following as simpler as possible?
Let $M$ be family of $n$ finite sets:
$$M = \{X_1, X_2, ..., X_n\}$$
Then 
$$\bigr(\forall k = 1, 2, ..., n : |X_{i_{1}}\cup X_{i_{2}}\cup\cdots\cup X_{i_{k}}|\geq k\bigl)\Rightarrow\bigl(\exists \chi_1,\chi_2,...\chi_n : \chi_i\neq\chi_j,\,\chi_i\in X_i\bigr)$$
In other words
If union of any k sets has at least k elements then there exist n different elements which represent whole family M.

This is one of the formulations of the Hall's theorem, and yes, there is a simple proof. It proceeds by induction and starts by considering two cases:


*

*$$\bigr(\forall k \in \{1, 2, ..., n-1\} : |X_{i_{1}}\cup X_{i_{2}}\cup\cdots\cup X_{i_{k}}|> k\bigl)$$

*$$\bigr(\exists k \in \{1, 2, ..., n-1\} : |X_{i_{1}}\cup X_{i_{2}}\cup\cdots\cup X_{i_{k}}|= k\bigl)$$


The first case can be solved by arbitrary assignment and induction hypothesis. The second case splits the graph into two subsets that can be solved independently and again applies induction hypothesis. E.g., see this.
I hope this helps $\ddot\smile$
