I have recently been trying to extend my knowledge of complex analysis by expanding to Riemann surface theory. I had been recommended Forster's Lectures on Riemann Surfaces as a good reference to go through and have been doing some of the exercises. The most recent exercise I have attempted has had me scratching my head, namely exercise 29.1. Admittedly I do not have a lot of experience in dealing with vector bundles so I think my intuition is off about how to approach this problem.
First, a definition:
Let $X$ be a Riemann surface and $\pi: E \to X$ a holomorphic vector bundle of rank $n$ on $X$. A holomorphic subbundle $F \subseteq E$ of rank $k$ is a subset such that the following holds. For every $x \in X$ there exists a holomorphic local trivialisation $$h: E_U \to U \times \mathbb{C}^n, \qquad E_U = \pi^{-1}(U),$$ of $E$ with $x \in U$ such that $$h(F_U) = U \times (\mathbb{C}^k \times 0)$$ where $F_U = E_U \cap F$.
My problem is that given a meromorphic section $f$ of $E$, I need to show that there exists a unique subbundle $F$ of rank 1 such that $f$ is a meromorphic section of $F$. I do know from the previous part of the exercise that if I am given a holomorphic section that never vanishes, then I can construct a holomorphic subbundle of rank 1. To do this, I need only to look at this locally, and so then the problem becomes constructing $h$ as an invertible linear transformation sending the standard basis of $\mathbb{C}^n$ over $\mathbb{C}$ to a basis including the vector $(f_1, \dots, f_n)$, the local representation of $f$. To tackle my current problem, I was thinking that I should somehow cunningly choose an open neighbourhood of $X$ so that the problem becomes easier to deal with, and then somehow use this result to obtain a subbundle of rank 1. However, I am unsure how to make any of this precise nor do I know how to proceed to show that such a subbundle is unique.
Any guidance would be much appreciated but I would like to add that I cannot use very heavy or "sophisticated" machinery. In particular, I can't use anything beyond Riemann surface theory that has not been presented in Forster, because I believe he has made these exercises possible to be completed with general undergraduate complex analysis knowledge or from the material that he has presented. Thanks for all the help!