In Hartshorne, Chap. IV: Curves on page 319 I found this proof:
Proposition 4.6. Let $X$ be an elliptic curve over $k$, with char $k \neq 2$, and let $P_0 \in X$ be a given point. Then there is a closed immersion $X \to \mathbb{P}^2$ such that the image is the curve
$$y^2 = x(x - 1)(x - \lambda)$$
for some $\lambda \in k$, and the point $P_0$ goes to the point at infinity $(0,1,0)$ on the $y$-axis. Furthermore, this $\lambda$ is the same as the $\lambda$ defined earlier, up to an element of $\Sigma_3$ as in (4.5).
PROOF. We embed $X$ in $\mathbb{P}^2$ by the linear system $\vert 3P_0 \vert $, which gives a closed immersion (3.3.3). We choose our coordinates as follows. Think of the vector spaces $H^0(\mathcal{O}_X(n P_0 ))$ as contained in each other,
\begin{equation} k = H^0(\mathcal{O}_X) \subset H^0(\mathcal{O}_X( P_0 )) \subset H^0(\mathcal{O}_X(2 P_0 )) \subset... \tag{1}\label{eq:0105star} \end{equation}
By Riemann-Roch, we have
$$\dim_k H^0(\mathcal{O}_X(n P_0 ))=n $$
for $n >0$. Choose...
Q: how the chain $$ k = H^0(\mathcal{O}_X) \subset H^0(\mathcal{O}_X( P_0 )) \subset H^0(\mathcal{O}_X(2 P_0 )) \subset... $$
is concretely constructed in the proof? to obtain an inclusion $H^0(\mathcal{O}_X( k P_0 )) \subset H^0(\mathcal{O}_X((k+1) P_0 ))$ we need an inclusion of $\mathcal{O}_X$-sheaves $\mathcal{O}(k P_0) \subset \mathcal{O}((k+1) P_0)$ but since $kP_0$ is nothing but a closed subscheme of $(k+1)P_0$. therefore wo have only a surjective map $(k+1)P_0 \to kP_0$ in "wrong" direction. consequently I don't know why the chain (1) make any sense.