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.