= Solution
For a nonzero <holomorphic vector bundle> $F$ on the curve, its <degree of a holomorphic vector bundle> and <slope of a holomorphic vector bundle> are
$$
\deg F=\deg\det F,\qquad \mu(F)=\frac{\deg F}{\operatorname{rank}F}.
$$
We obtain a bound independent of the particular <holomorphic subbundle> by fixing a <Hermitian metric on a holomorphic vector bundle> $E$ once and for all. Let $D_E$ be its <Chern connection>, $F_E=D_E^2$ its <Chern curvature>, and $\omega$ any positive area form on the <compact Riemann surface>. Write
$$
iF_E=K\omega.
$$
Here $K$ is a smooth Hermitian endomorphism of $E$. By <compactness>, its largest eigenvalue is bounded above by some finite constant $C$ on the whole curve.
Let $F\subseteq E$ be a <holomorphic subbundle> of positive rank $r$, with the induced metric. Use the smooth <orthogonal splitting of a vector subbundle> $E=F\oplus F^\perp$. In this splitting the connection has block form
$$
D_E=\begin{pmatrix}D_F&-B^\dagger\\B&D_Q\end{pmatrix},
$$
where $D_F$ is the <projected Chern connection> and $B$ is the <second fundamental form of a holomorphic subbundle>. The latter has type $(1,0)$: the $(0,1)$ part of $D_E$ preserves the holomorphic subbundle. The adjoint $B^\dagger$ includes conjugation of the form factor and has type $(0,1)$. Squaring this block matrix gives
$$
(F_E)_{11}=F_F-B^\dagger\wedge B,\qquad F_F=(F_E)_{11}+B^\dagger\wedge B.
$$
This derives the <curvature formula for a holomorphic subbundle> in the column-vector convention used here. Locally write $B=b\,dz$. Then
$$
B^\dagger\wedge B=-b^\dagger b\,dz\wedge d\bar z.
$$
The <degree of a holomorphic vector bundle> is the trace-curvature integral, since the trace is the curvature of the <determinant line bundle>. Hence
$$
\deg F=\frac{i}{2\pi}\int_X\operatorname{tr}F_F
=\frac{1}{2\pi}\int_X\operatorname{tr}(P_FK|_F)\,\omega
-\frac{1}{2\pi}\int_X i\operatorname{tr}(b^\dagger b)\,dz\wedge d\bar z.
$$
The last integrand is a nonnegative real two-form and is independent of the local unitary frame and coordinate. Since $K\le C I_E$, the compressed trace is at most $rC$ pointwise. Therefore
$$
\deg F\le\frac{rC}{2\pi}\int_X\omega,
\qquad
\boxed{\mu(F)\le\frac{C}{2\pi}\int_X\omega.}
$$
This is the <subbundle slope bound from ambient Chern curvature>. The right side depends only on $E$ and the initially chosen metrics, and is finite. It proves the requested uniform upper bound for every positive-rank subbundle, without assuming any stability property of $E$ or using a bound that depends on $F$.
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