If a $||f_n-f||_p \to 0$ then $f_n \to f$ uniformly a.e.?

Question

While trying to prove Hardy's inequality, I showed that if $F(x)=\frac 1x \int_0^x f(t)\mathrm{d}t$ then $||F||_p \le \frac {p}{p-1} ||f||_p$ for any $f\in C_c(0,\infty)$ and $1\le p < \infty$ . Now, since $C_c(0,\infty)$ is dense in $L^p(\mu)$ that means that for any function $f$ in $L^p$ there exists a sequence of functions $f_n \in C_c$ such that $||f_n-f||_p\to 0$. This is good, since then $$\lim ||F_n||_p=||F||_p \le \frac {p}{p-1} ||f||_p=\lim \frac {p}{p-1} ||f_n||_p$$ The only problem I have with this is that for the above to work, I need to be able to move the limits into the integrals, which can only be done (as far as the book has showed) through the Monotone/Dominated Convergence Theorems and uniform convergence of $f_n$. Since I have no clear way of bounding the $f_n$ so as to satisfy the first two theorems' conditions, I went for showing that $f_n$ converges uniformly.
We are working in $L^p$ space. If $f_n$ does not converge a.e. uniformly to $f$, then there exist $\epsilon > 0$ and $X$ such that for all $n$, $|f_n-f| > \epsilon$ on $X$, which implies that $\int_X |f_n-f|^p > \epsilon^p\mu (X)>0$ and since $$\left(\int_{\Omega} |f_n-f|^p\mathrm{d}\mu\right)^{\frac 1p}=\left(\int_{X} |f_n-f|^p\mathrm{d}\mu+\int_{X^c} |f_n-f|^p\mathrm{d}\mu\right)^{\frac 1p} >\epsilon \mu(X)^{\frac 1p}$$ Then $||f_n-f||_p$ is bounded below by a non-zero number and therefore cannot get arbitrarily close to zero.
First, I do not know if this is correct, since all of the solutions I have read so far do not use this argument (they use something called Fubini's theorem, which is not presented until further in the book), so I was wondering if this proof is incorrect/incomplete, and if so, how to find the right solution without using Fubini?