I want to show that there exist an $\epsilon \geq 0$ $$ \max_{z \in C(0,1)} \left| p(z)-\frac{1}{z}\right|\geq \epsilon $$ for every polynomial $p(z)$, where $C(0,1)$ denotes the unit circle.
Approach: I tried a proof by contradiction. So suppose $$ \max_{z \in C(0,1)} \left| p(z)-\frac{1}{z}\right|= 0. $$ Then $|zp(z)-1|=0$ on $C(0,1)$. According to maximum modulus principle $zp(z)-1$ is constant (since $zp(z)-1$ cannot attain a maximum in the unit disc $D(0,1)$). Hence $p(z)=1/z$ on the punctured plane, contradiction. Is this the correct reasoning? Thanks in advance.