Say I have a finite set $A$ and know its cardinality. How can I prove that, by repeatedly applying some algorithm, which removes a known number of elements from $A$ each time, its cardinality will stop at some value? It should stop because the aforementioned algorithm requires a fixed number of elements present in the set.
I tried induction, but couldn't figure out a condition over $n \in \mathbb{N}$ that resembled this problem. I also thought about limits of sequences, but this would be a finite sequence.
Any hint is appreciated! Thanks in advance.