Equivalence of Definitions of Meromorphic Function

$(1)$ implies $(2)$
Let $f$ be a meromorphic function by definition $1$.

By Meromorphic Function is Quotient of Holomorphic Functions:
 * $f$ can be expressed as the ratio of two holomorphic functions.

Thus $f$ is a meromorphic function by definition $2$.

$(2)$ implies $(1)$
Let $f$ be a meromorphic function by definition $2$.

Thus $f$ is a meromorphic function by definition $1$.