Here is the question:
Correcting the calculation of a norm and continuity of a sequence of functions.
in the solution given by infinity I do not understand this step in the line before last the last part.
$|\int_0^1g_n h|\to \int_0^1|h|$$
Could anyone explain this for me please? does this is because $$|\int_0^1g_n h | \leq \int_0^1 \left\lvert g_n h \right\rvert \leq \lVert g_n \rVert_ {\max} \int_0^1 | h| \leq \lVert h \rVert_1 \quad \text{as} \quad \|g_{n}\|_{max} \leq 1~?$$
Since $g_n\to sgn(h)$ in $L^1$, we may (by picking a subsequence) suppose without loss of generality that $g_n \to sgn(h)$ almost everywhere. Note also that $|g_n h|\leq |h|$. By the dominated convergence theorem, $$ \lim_{n\to\infty}\int_{[0,1]} g_n h =\int_{[0,1]}\lim_{n\to\infty} g_n h =\int_{[0,1]}sgn(h) h =\int_{[0,1]} |h|. $$