I am confused about what can and cannot be done in a proof in order to avoid having an infinite number of "steps", thus making the argument invalid.
For instance, let $A_{0}, A_{1}, \dotsc$ be subsets of $\mathbb{R}^{n}$, and suppose that I want to show that $A_{0}$ satisfies some desired property after shrinking $A_{1}, A_{2}, \dotsc$.
Would this be OK? Or would it involve infinitely many "steps"?
In general, how can I make sure my proofs are free from this kind of problems?
EDIT: Using induction, suppose that I can prove that if I shrink $A_{1}, A_{2}, \dotsc$, then $A_{0}$ satisfies the desired property. In other words, I can show that my statement holds when I have just two subsets, and that if it holds for $n$ subsets, then it also holds for $n+1$. But how is this supposed to help me? The hypothesis requires me to shrink an infinite amount of subsets, which looks like too big a task.