How to prove that there is no non constant analytic function $f$ on open unit disc with center at zero such that
$$|f(\frac{1}{n})|<\frac{1}{2^n}, n\in\mathbb N, n\geq 2$$
I tried by Identity theorem but fail to apply it as function is not given exact equal to another value . By given conditions I only know that $f(z)=z^mg(z)$ type function. Then $$|2^mf(\frac{1}{n})|< |g(\frac{1}{n})| $$ Now I am stuck. Thank you.