I want to prove the following:
Let $A$ be a ring and $n$ a natural number. If the left $A$-module $A^n$ contains a free subset of $n+1$ elements, then $A^n$ already contains an infinite free subset.
Since we can embed $A^{n+1}$ into $A^n$, we can embed $A^{n+2}$ into $A^{n+1}$, and so on. So $A^n$ contains finite free subsets of all sizes, but I really have no clue how to construct an infinite free subset from that.
Can someone help me?