Well, I know that I'm going to have some vote-down, because this should be a very simple property but the real story is that... I am not able to prove it!
For any fixed $n\in\mathbb N$, I have a finite partition of the natural numbers
$$ \mathbb N=A_1^n\cup...\cup A_{k(n)}^n $$
(every set of the partition is infinite). Suppose that the sequence $k(n)$ is bounded. I want to make a choice of a set $A_{i_n}^n$ for each partition such that $A=\bigcap_{n=1}^\infty A_{i_n}^n$ is infinite.
Intuitively this should be possible, and I have a sketch of the proof making use of the upper density on natural numbers, but I don't think it is completely right and anyway it sounds a bit too complicated.
Motivation: I am basically extracting infinite subsequences from a sequence and I want to be sure to find, at the end of the story, a subsequence.