Stalk of the direct image of a locally free sheaf

Question

Suppose $C$ is an irreducible algebraic curve, $U \subset C$ is an open subset, $M$ is a locally free $\mathcal{O}_X$-module of rank $\mathrm{n}$. Let $j : U \rightarrow C$ be the open embedding, I want to calculate $(j_{*}M \vert_U)_p$ for $p$ a point on the boundary of $U$, and I guess it is $K_{C,p} \otimes_{\mathcal{O}_{C,p}} M_p$, where $\mathcal{O}_{C,p}$ is the stalk of the structure sheaf of $C$ at $p$, and $K_{C,p}$ is its quotient field. My proof works as follows, pick $p \in W \subset C$ such that $M \vert_W \simeq \mathcal{O}_X^{n}$, then $$ (j_{*}M \vert_U)_p = \varinjlim_{V \ni p} M(U \cap V) = \varinjlim_{V \ni p, V \subset W} M(U \cap V) \simeq \varinjlim_{V \ni p, V \subset W} \mathcal{O}_W^n (U \cap V) = K_{C,p}^n \simeq K_{C,p} \otimes_{\mathcal{O}_{C,p}} M_p $$ where the first equality comes from the fact that being $W$ dense it intersects every open non-empty subset, hence we can calculate the colimit on open subsets contained in $W$ and then use restriction maps; the second isomorphism comes from the choice of $W$; the third equality comes from the fact that $U \cap V$ doesn't contain $p$, hence $\mathcal{O}^n_{W}(U \cap V)$ contains $x_p^{-1}$ for each $V$, where $x_p$ is a local coordinate at $p$. Is my proof correct?