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    $\begingroup$ You had a typo in the first underbrace -- which should be $g(E(X)) - E(g(X))$. By the way, I think your proof is already clear and straightforward enough (as well as the underlying idea), hence further essential reduction seems unlikely. $\endgroup$
    – Zhanxiong
    Commented Sep 16, 2023 at 2:59
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    $\begingroup$ To the closevoter: This seems eminently on-topic to me! $\endgroup$
    – kjetil b halvorsen ♦
    Commented Sep 16, 2023 at 12:23
  • $\begingroup$ @Zhanxiong : typo fixed $\endgroup$
    – Michael Hardy
    Commented Sep 16, 2023 at 21:22