This result appears to be ubiquitous as an algebra exercise. How do you prove this result?

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> Let $A$ be an integral domain with field of fractions $K$, and let $A\_{\mathfrak{m}}$ denote the localisation of $A$ at a maximal ideal $\mathfrak{m}$ considered as a subring of $K$. Prove that $$A = \bigcap\_{\mathfrak{m}} A\_{\mathfrak{m}}\,,$$ where the intersection is taken over all maximal ideals $\mathfrak{m}$ of $A$.
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