I'm posting this question just to be sure if my solution is correct or not.
I have a sequence $\{f_{n}\}$, where each $f_{n}:\mathbb{R}\to\mathbb{R}$ is continuous on $\mathbb{R}$ and $|f_{n}(x)|< a_{n}$ for all $n=1,2,3,\dots$ , for some sequence $a_{n}$ converges to 0. I'm trying to prove that $\frac{|f_{n}(x)|}{a_{n}}\to 0$ as $n\to\infty$.
I started with: since $|f_{n}(x)|\to 0$ as $n\to\infty$ (because $a_{n}\to 0$) for all $x\in\mathbb{R}$, this means that:
for any $\epsilon>0$ there exists $N_{\epsilon}>0$ such that $|f_{n}(x)|< \epsilon$ for all $n> N_{\epsilon}$ and all $x\in \mathbb{R}$, (uniform convergence)
Also, we can say: for any $j\in \mathbb{N}$ (by taking $\epsilon=a_{j}^{2}$) there exists $N_{j}>0$ such that $|f_{n}(x)|< a_{j}^{2}$ for all $n> N_{j}$ and all $x\in \mathbb{R}$, therefore: $\frac{|f_{n}(x)|}{a_{n}}<\frac{a_{j}^{2}}{a_{n}} \to 0,\; as \;j\to \infty$ for all $n>N_{j}$ and all $x$.
So, for large $n$ we have $\frac{|f_{n}(x)|}{a_{n}}\to 0$ by taking the limit as $j\to \infty$.
I think there is something wrong somewhere in my argument, if so please correct me!