Gromov compactness implies properness

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TODO: As claimed in regularized moduli spaces, show that Gromov compactness can be formulated as saying that for any w_{0}\in \mathbb{R} the restricted section \overline \partial _{{J,Y}}|_{{{\mathcal  {X}}_{{w_{0}}}(\underline {x})}}:{\mathcal  {X}}_{{w_{0}}}(\underline {x})\to {\mathcal  {Y}}_{J}(\underline {x})|_{{{\mathcal  {X}}_{{w_{0}}}(\underline {x})}}:=\pi ^{{-1}}{\bigl (}{\mathcal  {X}}_{{w_{0}}}(\underline {x}){\bigr )} is proper.

Recall from [Definition 4.1 HWZ-III] that a section being proper requires the zero set \overline \partial _{{J,Y}}^{{-1}}(0)\cap {\mathcal  {X}}_{{w_{0}}}(\underline {x}) to be compact.