IMC 2011 Problem 4

Let $A_1,A_2,\dots, A_n$ be finite, nonempty sets. Define the function $$f(t)=\sum_{k=1}^n \sum_{1\leq i_1<i_2<\dots<i_k\leq n} (-1)^{k-1}t^{|A_{i_1}\cup A_{i_2}\cup \dots\cup A_{i_k}|}.$$ Prove that $f$ is nondecreasing on $[0,1].$ ( $|A|$ denotes the number of elements in $A.$ )