injective morphism between line bundles on curves

Question

Let $X$ be a smooth projective, irreducible, curve, $\mathcal{L}$ be an invertible sheaf on $X$ and $\mathcal{L}' \subset \mathcal{L}$, an invertible subsheaf. Is $\deg(\mathcal{L}') \le \deg(\mathcal{L})$?

Answer 1

Yes. Twist the inclusion by $\mathcal{L}^{-1}$, so that $\mathcal{L}^{-1} \otimes \mathcal{L}' \subset \mathcal{O}_X$, in particular it is an ideal sheaf (hence has non-positive degree).