Fact:If $ X $ and $ Y $ are varieties and we have $ \mathcal{O}_{X,q} \cong \mathcal{O}_{Y,q} $ then there are neighborhoods $U$ of $p$ and $V$ of $q$ which are isomorphic.

I understand the intuition behind this result: Since the open sets are so big, the scheme $$ \varprojlim_{p \in U\subseteq X \; \text{open}} U $$ captures a lot of infomation about $X$ at the point $p$.

I can imagine a proof which goes like this: We can assume without loss of generality that the varieties in question are affine. Therefore we are reduced to proving that an isomorphism $ R_{\mathfrak{p}} \to S_{\mathfrak{q}}$ of rings induces an isomorphism $ R_f \to S_g$ for suitable elements $f \notin \mathfrak{p} $ and $ g \notin \mathfrak{q} $.

This approach is a bit disappointing in my opinion because the result is so geometric and we reduce it to a formal proof about localization.

Question:Can anyone explain a geometric proof of this result to me?