Any degree $2$ divisor on genus $1$ curve moves in a pencil

Question

I have a genus $1$ curve, and would like to prove a fact from the headline:

Let $C$ be a genus $1$ curve and let $D$ be a degree $2$ divisor. Then $l(D)=2$.

Is this possible to prove this using Riemann-Roch only, without any further assumptions on the divisor $D$?

Answer 1

If $g=1$ Riemann-Roch tells us $l(D)-i(D)=\deg D$. If $\deg D>0$ then Serre's duality $i(D)=l(-D)$ and vanishing of $l(D)$ for divisors of negative degree give us $l(D)=\deg D$.