For $f \in L^1\left(\mathbb{R}^N\right)$ and let $g \in L^p\left(\mathbb{R}^N\right)$ with $1 \leq p \leq \infty$, we define $$ (f \star g)(x)=\int_{\mathbb{R}^N} f(x-y) g(y) d y. $$
A sequence of mollifiers $\left(\rho_n\right)_{n \geq 1}$ is any sequence of functions on $\mathbb{R}^N$ such that $$ \rho_n \in C_c^{\infty}\left(\mathbb{R}^N\right), \quad \operatorname{supp} \rho_n \subset \overline{B(0,1 / n)}, \quad \int \rho_n=1, \rho_n \geq 0 \text { on } \mathbb{R}^N . $$ In what follows we shall systematically use the notation $\left(\rho_n\right)$ to denote a sequence of mollifiers. I'm reading the proof of below theorem in Brezis's Functional Analysis, i.e.,
Proposition 4.21. Assume $f \in C\left(\mathbb{R}^N\right)$. Then $\left(\rho_n \star f\right) \underset{n \rightarrow \infty}{\longrightarrow} f$ uniformly on compact sets of $\mathbb{R}^N$.
It seems from the logic of the proof that there is a typo. I think the author meant $f \star \rho_n$ rather than $\rho_n \star f$.
Could you confirm if my understanding is correct?
Proof. Let $K \subset \mathbb{R}^N$ be a fixed compact set. Given $\varepsilon>0$ there exists $\delta>0$ (depending on $K$ and $\varepsilon$ ) such that $$ |f(x-y)-f(x)|<\varepsilon \quad \forall x \in K, \quad \forall y \in B(0, \delta) . $$ We have, for $x \in \mathbb{R}^N$, $$ \begin{aligned} \left(\rho_n \star f\right)(x)-f(x) & =\int[f(x-y)-f(x)] \rho_n(y) d y \\ & =\int_{B(0,1 / n)}[f(x-y)-f(x)] \rho_n(y) d y . \end{aligned} $$ For $n>1 / \delta$ and $x \in K$ we obtain $$ \left|\left(\rho_n \star f\right)(x)-f(x)\right| \leq \varepsilon \int \rho_n=\varepsilon . $$