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The flat unitary connection on $E_V$ is its Chern connection, since it preserves the induced Hermitian metric and its $(0,1)$ part is the Dolbeault operator. Its curvature is zero. The trace-curvature formula for the degree of a holomorphic vector bundle therefore givesFor any positive-rank proper holomorphic subbundle $F\subset E_V$, equip $F$ with the induced metric and use the block-connection calculation of Question 4(a). Since the ambient curvature vanishes, its Chern curvature iswhere $B=bdz$ is the second fundamental form of a holomorphic subbundle. ConsequentlyThe term being integrated is nonnegative, so its sign gives the inequality for every such subbundle, not just for parallel subbundles. By the definition of a semistable holomorphic vector bundle,If semistability is formulated using coherent subsheaves instead, saturate a subsheaf on the smooth curve. The saturation has the same rank, at least as large a degree, and is a subbundle because a torsion-free sheaf on a smooth curve is locally free. The inequality for the saturated subbundle therefore also bounds the original subsheaf. Finally, equality for a subbundle forces $B=0$, so it is parallel; invariant subspaces of $V$ correspond to these degree-zero parallel subbundles. Semistability is the conclusion required here, rather than stability for every unitary representation.

The slope of a nonzero holomorphic vector bundle on a compact Riemann surface is its degree of a holomorphic vector bundle divided by its rank of a vector bundle. It measures degree per fibre dimension. Comparing subbundle slopes defines semistable holomorphic vector bundles.