I have a homework problem I am unable to finish. Could someone please help me proceed?
Problem: Suppose $(X, \mathcal{A}, \mu)$ is a finite measure space (that is, $\mu(x) < \infty$ ) and $f:\mathbb{R} \to \mathbb{R}$ is bounded and continuous. Define a composition mapping $ F (u) = f \circ u$. Show that if $1\leq p < \infty$ , the map $ F: L^p(X) \to L^p(X)$ is continuous.
My working: Let $(u_n)_{n\in \mathbb{N}}$ be a sequence of functions such that $ \| u_n - u \|_p \to 0.$ We need to show that $ \| F(u_n) - F(u) \|_p \to 0.$ Since $f$ is bounded, there exists $M\in \mathbb{R}$ such that $|f(x)| < M $ for all $x\in \mathbb{R}$. Thus a dominating function of $h_n(x) = | f( u_n(x) ) - f(u(x))|^p $ is $g(x) = (2M)^p $ and since $ \int_X (2M)^p dx = (2M)^p \int_X d\mu = (2M)^p \mu(X) < \infty $, by the Dominated Convergence Theorem we have that $ \lim_{n\to\infty} \int_X | f( u_n(x) ) - f(u(x))|^p dx = \int_X \lim_{n\to\infty} | f( u_n(x) ) - f(u(x))|^p dx. $ By the continuity of $f$, the absolute value function and exponentiation by $p$, we can then get $ \lim_{n\to\infty} \int_X | f( u_n(x) ) - f(u(x))|^p dx = \int_X | f( \lim_{n\to\infty} u_n(x) ) - f(u(x))|^p dx. $ Now, from here I am unsure how to proceed. One thing I know is that if $(u_n)_{n\in \mathbb{N}}$ is a sequence of functions such that $ \| u_n - u \|_p \to 0$, then there exists a subsequence $(u_{n_k}) $ such that $u_{n_k}$ converges pointwise to $u$ almost everywhere. If we let $N$ denote the set where the pointwise convergence does not hold, then we have $ \int_X| f( \lim_{n\to\infty} u_{n_k}(x) ) - f(u(x))|^p dx = \int_{X/N} 0 dx + \int_N | f( \lim_{n\to\infty} u_{n_k} (x) ) - f(u(x))|^p dx =0$ where the first integral is $0$ because the integrand is $0$, and the second integral is zero because $\mu(N) = 0$. So that shows the limit I want for at least a subsequence, but I don't know how to connect the others back to it. Any help would be greatly appreciated, but I would especially prefer if your answer finishes off my proof leaving most of it intact, rather than perhaps an entirely different start. Thank you.