From Folland's Real Analysis: Modern Techniques and Their Applications there is the following result:
Folland, Theorem 8.14, (a): Suppose that $\phi\in L^1$ and $\int\phi(x)\text{d}x=a$. If $f\in L^p$ for $1\le p<\infty$ then $f*\phi_t\to af$ in $\|\cdot\|_p$ as $t\to\infty$.
Here, $\phi_t(x)=t^{-n}\phi(t^{-1}x)$. The proof of a variant of this problem is asked and discussed here: Why convolution of integrable function f with some sequence tends to f a.e.
I am interested in another variant of this problem, and have followed the proof outlined in Folland, but am now at a point where I am not sure how to continue. I feel that I am very close to drawing out the result I need.
In my variation of the problem, I have $\phi_n(x):=n\phi(nx)$ for $n\in\mathbb N$ where $\phi\in C^\infty_c(\mathbb R)$, $\phi(x)\ge0$ for all $x\in\mathbb R$, the support of $\phi$ is contained in $[-1,1]$ and $\int_\mathbb{R}\phi(x)\text{d}x=1$. I am concerned with the particular case when $p=2$
Here is my attempt so far. We want to show that $\lim_{n\to\infty}\|f*\phi_n-f\|_2=0$, so begin by considering what $\|f*\phi_n-f\|_2$ looks like. We obtain, $$\|(f*\phi_n)(x)-f(x)\|_2=\left(\int_{\mathbb R}\left|\int_\mathbb{R} n\phi(ny)\left[\tau_y\left(f(x)\right)-f(x)\right])\,\text{d}y\right|^2\,\text{d}x\right)^{1/2},$$ where $\tau_y$ denotes the translation of $f$ by $y$, i.e. $\tau_y\left(f(x)\right)=f(x-y)$. By Minkowski's integral inequality, we obtain that, $$\left(\int_{\mathbb R}\left|\int_\mathbb{R} n\phi(ny)\left[\tau_y\left(f(x)\right)-f(x)\right])\,\text{d}y\right|^2\,\text{d}x\right)^{1/2}\le\int_\mathbb{R}|n\phi(ny)|\|\tau_y\left(f(x)\right)-f(x)\|_2\text{d}y\le 2\|f\|_2,$$
where we have made use of the fact that $\|\tau_y\left(f(x)\right)-f(x)\|_2\le2\|f(x)\|_2$. Note that this leaves only $\int_\mathbb{R}|n\phi(ny)|\text{d}y$, which we can integrate with the substitution $y=s/n$ to see that this integral is 1.
Putting this all together, we now have that, $$\lim_{n\to\infty}\|f*\phi_n-f\|_2\le2\|f(x)\|_2.$$ I don't think that I can apply LDCT because of the factor of $n$ in $n\int_\mathbb{R}\phi(ny)\text{d}y$. Although, we do have at least that the limit of the integral of this sequence is bounded. Can we work with this somehow to draw out the result we need? Or is this the incorrect way to go about the problem?