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Use the defining criterion that a flat module $E$ makes $u\otimes 1_E$ injective for every injective module homomorphism $u$. By right exactness of the tensor product, preserving injections is equivalent to preserving every short exact sequence.

First assume $E$ is flat over $A$, and fix a maximal ideal $\mathfrak{m}$. Let $u:U\hookrightarrow V$ be an injection of $A_{\mathfrak{m}}$-modules. Regard it as an injection of $A$-modules. There are natural module isomorphismsFor example, $u_0\otimes (e/s)$ maps to $s^{-1}{u}_0\otimes e$, with inverse $u_0\otimes e\mapsto u_0\otimes (e/1)$. These are well-defined because every $s\notin \mathfrak{m}$ already acts invertibly on $U$. Under these isomorphisms the tensor map is $u{\otimes }_A{1}_E$, which is injective by flatness. Thus $E_{\mathfrak{m}}$ is flat over $A_{\mathfrak{m}}$.

For the converse, assume every $E_{\mathfrak{m}}$ is flat over $A_{\mathfrak{m}}$ and take any injection $u:U\hookrightarrow V$ of $A$-modules. By part (c), each $u_{\mathfrak{m}}$ is injective. Since localization commutes with tensor products, the localization of $u{\otimes }_A{1}_E$ identifies withIt is injective by the assumed local flatness. Apply part (c) again, now to $u{\otimes }_A{1}_E$, to conclude that it is injective globally. Since $u$ was arbitrary, $E$ is flat. ThereforeThis proves flatness is local without any finite-generation hypothesis.