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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.)

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up vote 2 down vote accepted

$|x|\leqslant|x-c|+|c|\lt1+|c|$ and $|c|\lt1+|c|$ hence $$|x|^{n-1-k}|c|^k\lt(1+|c|)^{n-1-k}(1+|c|)^k=(1+|c|)^{n-1}\leqslant(1+|c|)^n.$$

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