I hate to ask this but I always struggle with inequalities:
suppose $c \in \mathbb{R}$. how can I show that if $| x - c| < 1$ then $|x|^{n-1-k}|c|^k < (1 + |c|)^n$ for each $k = 0,\dots,n-1$ ?
hints are totally enough, I should be able to work out the details myself. thanks a lot !! (P.S. not a hw question, I am working through a study guide where this is a statement.)