I saw a statement in a question Non-compact Kähler manifolds which admit a positive line bundle, which says that any compact complex manifold that admits a positive line bundle must be a Kähler manifold automatically. Can someone explain to me why?

The second confusion appears in J.P. Demailly's book "Analytic Methods in Algebraic Geometry", Chapter 8, Corollary 8.3 here. How can he directly prove a compact complex manifold carrying a holomorphic big line bundle is a Moishezon manifold? There's not any hints and proof below Corollary 8.3 doesn't cover this statement.

(This is the first time I ask question on MathOverflow, I hope I can express my question accurately enough, and show my great gratitude to anyone who is willing to answer my question.)

oneanswer as accepted, and it is easier to do so when there is only one question to answer.) $\endgroup$