\(\displaystyle |P(x)|\leq C|1+|x|+|x|^{2}+..x^{n}|=C\frac{1-|x|^{n+1}}{1-|x|}\\

||x|-|a||\leq |x-a|<1\\

|a|-1\leq |x|\leq |a|+1\\

\)

\(\displaystyle \frac{1}{1-|x|}\leq- \frac{1}{|a|}\\\)

\(\displaystyle \frac{1-|x|^{n+1}}{1-|x|}\leq \frac{1}{|a|}(-1+|x|^{n+1})\\\\\)

\(\displaystyle |x|^{n+1}\leq (1+|a|)^{n+1}=1+(n+1)|a|+\frac{1}{2}(n+1)n|a|^{2}+...\\\\

\leq 1+(n+1)|a|(1+|a|)^{n}\\\\

\)

\(\displaystyle \frac{1-|x|^{n+1}}{1-|x|}\leq (n+1)(1+|a|)^{n}\)

Came back to see who actually thanked me and caught two typo's, one trivial and one that should be corrected.

The trivial one is a missing C.

In a few places 1/(1-x) should be 1/(1-|x|)

The corrections are made in the above quote.

Thanks Ould Youbba. Without your thanks I would have missed the typo's

I originally did the whole thing in plain text and then resubmitted it in Latex to satisfy skipjack. Subsequently skipjack deleted the plain text version.