Show that for any polynomial $p(z)$ there is a $z$ with $|z|=1$ such that $|p(z)-1/z|\geq 1$. I'm having a bit of trouble on another problem, and I'm not sure where to start:
Show that for any polynomial $p(z)$ there is a $z$ with $|z|=1$ such that $|p(z)-1/z|\geq 1$. 
Could anybody get me started with a tip or two? Thanks in advance.

You have to show that you can find $z$ of modulus $1$ such that $|zp(z)-1|\geq 1$.If it's not the the case, you apply Rouché's theorem to get a contradiction.
