Equality of two analytic functions based on a condition

Question

Suppose that $f$ and $g$ are two analytic functions on the set $\phi$ of all complex numbers with $f(\frac{1}{n})=g(\frac{1}{n})$ for $n=1,2,3,\ldots$ Then show that $f(z)=g(z)$ for each $z$ in $\phi$.

I am not able to even start proving it. Can some one please help me?

Answer 1

Hint: Check the identity theorem for analytic functions:

If two functions are analytic in a region $G$, and if they coincide in a neighborhood, however small, of a point $z_0$ of $G$, or only along a path segment, however small, of a point $z_0$ of $G$, or also only for an infinite number of distinct points with the limit point $z_0$, then the two functions are equal everywhere in $G$.