Degree of ample bundle over projective curve is positive
Solution 1:
The result and your argument (which is the first I would think of) are still valid for integral projective curves, but the theory is more delicate. For example, you can find the result as 7.3.2 Proposition 3.25(b) of Liu's wonderful book, in which he develops Riemann-Roch results with minimal assumptions on what constitutes a "curve" (e.g. possibly singular, nonreduced).