Sequence of function depending on $n$ and $\epsilon$

Suppose that $\{f_n\}$ be a sequence of functions in $\Bbb R$ that converges to $f(x)$. Then can we write the following ?:

"For every $\epsilon >0$, there exists a natural number $N$ such that $\displaystyle |f_n(x)-f(x)|<\frac{\epsilon}n$ for all $n\ge N$ and for all $x\in \Bbb R$."

Is it correct? If not then can I expect a brief explanation? may be by an example.

Your sentence Suppose that $\{f_n\}$ be a sequence of functions in $\mathbb R$ that converges to $f(x)$ is very confusing.

A sequence of functions can not converge to a number. Hence either you speak of pointwise convergence meaning that for each $x \in \mathbb R$ $\{f_n(x)\}$ converges to $f(x)$. Or you're speaking of uniform converge.

You need to chose your battle!