The lemma and a part of its proof are given below:
My question is:
Why $h_{n}(x) \rightarrow h(x)$? could anyone explain this for me, please?
2026-04-03 02:38:01.1775183881
Elaboration of a step in the proof of Fatou's Lemma (Royden $4^{th}$ edition pg.82).
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Because $h \le f$ then for any $x$ then $h(x)=f(x)$ or $h(x)<f(x)$. The convergence in the first case is clear from the definition of $h_n$, now suppose that $h(x)<f(x)$, then from the definition of $h_n$ we have that $h_n(x)=h(x)$ for all $n\ge N$ for some large enough $N\in\Bbb N$.