I have the following exercise to do: For the events $A_i \in \mathfrak{C}, i \in \mathbb{N}$ of a probability space $(\Omega,\mathfrak{C},\textbf{P})$ prove the Inequality of Kounias
$$\textbf{P}(\bigcup_{i=1}^{n}A_i)\leq min_{k\in{\{1,...,n\}}}\{\sum_{i=1}^n\textbf{P}(A_i)-\sum_{i:i\neq k}\textbf{P}(A_i\cap A_k)\}$$
I tried to use the properties of the probability measure $\textbf{P}$ and the $\sigma$-Algebra $\mathfrak{C}$ but I'm stuck because I don't know anything about the events $A_i$.
Fix a $k$, then you can write: $$ \mathbb{P}(\cup_{i = 1}^n A_i) = \mathbb{P} (A_k \bigcup_{i \neq k} (A_i \setminus A_k)) \le \mathbb{P} (A_k) + \sum_{i \neq k} \mathbb{P}(A_i \setminus A_k) $$ Now since $\mathbb{P}(A_i \setminus A_k) = \mathbb{P}(A_i) - \mathbb{P}(A_k \cap A_i)$ you get the desired result.