The degree of the minimal polynomial decreases to the limit

I bet this works: First show that (in any normed vector space) if $x_1,\dots,x_n$ are independent then there exists $\epsilon>0$ such that if $||x_j-y_j||<\epsilon$ for all $j$ then $y_1,\dots,y_n$ are independent. Apply that to $I,A,A^2,\dots$.