By local nonsatiation there exists a sequence $x_n\rightarrow x$ with $x_n\succ x\,\,\forall n$. By transitivity $x_n\succ x^*\,\,\forall n$. By contraposition $p\cdot x_n>m\,\,\forall n$. By continuity of the dot product $p\cdot x\ge m$. This is pretty much a somewhat more compact version of your own approach.

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