= Solution
A <cusp form> is <holomorphic> in the upper half-plane by definition, so it is <meromorphic> there. In the reverse direction suppose $f$ is <meromorphic> and
$$
|f(\tau)|y^{k/2}\leq B\qquad(\tau\in\mathbb H).
$$
On a small closed disk around an interior point, $y$ is bounded above and away from zero. Hence $|f|$ is bounded there by a fixed constant. A <pole> of positive order would make $|f|$ unbounded along points approaching its center, a contradiction. Every possible <pole> is therefore removable, and \b[$f$ is <holomorphic> on $\mathbb H$]. Meromorphicity alone would not establish this conclusion; the bounded invariant <norm> is being used here.
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